https://www.mathepedia.wiki/w/api.php?action=feedcontributions&feedformat=atom&user=InfernalAtom683Mathepedia - User contributions [en]2026-05-21T18:13:15ZUser contributionsMediaWiki 1.45.1https://www.mathepedia.wiki/w/index.php?title=Seifert-Van_Kampen_theorem&diff=116Seifert-Van Kampen theorem2026-05-14T10:54:34Z<p>InfernalAtom683: </p>
<hr />
<div>== Statement ==<br />
Let <math>X</math> be a topological space and let <math>\mathcal{U}=\{U_i\}_{i\in I}</math> be an open cover of <math>X</math>.<br />
<br />
Let <math>\mathcal{F}</math> be the set of all finite non-empty intersections of members of <math>\mathcal{U}</math>:<br />
<math display="block">\mathcal{F}=\left\{\left.\bigcap_{i\in J}U_i\right|\emptyset\neq J\subseteq I, |J|<\infty\right\}</math>.<br />
<br />
Regard <math>\mathcal{F}</math> as a category whose objects are the elements of <math>\mathcal{F}</math> and in which there is a unique morphism <math>V\to W</math> whenever <math>V\subseteq W</math>.<br />
<br />
Let <math>\Pi_1\colon \mathcal{F}\to \mathsf{Gpd}</math> be the functor that sends each <math>V\in \mathcal{F}</math> to its fundamental groupoid <math>\Pi_1(V)</math> and each inclusion <math>V\hookrightarrow W</math> to the induced morphism of groupoids.<br />
<br />
Then the canonical morphism <br />
<math display="block">\operatorname{colim}_{V\in \mathcal{F}}\Pi_1(V)\to \Pi_1(X)</math><br />
is an isomorphism of groupoids.</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Seifert-Van_Kampen_theorem&diff=115Seifert-Van Kampen theorem2026-05-14T10:52:25Z<p>InfernalAtom683: </p>
<hr />
<div>== Statement ==<br />
Let <math>X</math> be a topological space and let <math>\mathcal{U}=\{U_i\}_{i\in I}</math> be an open cover of <math>X</math>.<br />
<br />
Let <math>\mathcal{F}</math> be the set of all finite non-empty intersections of members of <math>\mathcal{U}</math>:<br />
<math display="block">\mathcal{F}=\left\{\left.\bigcap_{i\in J}U_i\right|\emptyset\neq J\subseteq I, |J|<\infty\right\}</math>.<br />
<br />
Regard <math>\mathcal{F}</math> as a category whose objects are the elements of <math>\mathcal{F}</math> and in which there is a unique morphism <math>V\to W</math> whenever <math>V\subseteq W</math>.<br />
<br />
Let <math>\Pi_1\colon \mathcal{F}\to \mathsf{Gpd}</math> be the functor that sends each <math>V\in \mathcal{F}</math> to its fundamental groupoid <math>\Pi_1(V)</math> and each inclusion <math>V\hookrightarrow W</math> to the induced morphism of groupoids.<br />
<br />
Then the canonical morphism <br />
<math display="block">\colim_{V\in \mathcal{F}}\Pi_1(V)\to \Pi_1(X)</math><br />
is an isomorphism of groupoids.</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Seifert-Van_Kampen_theorem&diff=114Seifert-Van Kampen theorem2026-05-14T07:37:20Z<p>InfernalAtom683: Created page with "== Statement == Let <math>X</math> be a topological space, and <math>U, V\subset X</math> be open sets such that <math>X = U\cup V</math>, and <math>U</math>, <math>V</math> and <math>U\cap V</math> are path-connected. Take a basepoint <math>x_0\in U\cap V</math> with inclusion maps: <math display="block">i\colon U\cap V\hookrightarrow U,\quad j\colon U\cap V\hookrightarrow V,\quad k\colon U\hookrightarrow X,\quad l\colon V\hookrightarrow X,</math> then the following d..."</p>
<hr />
<div>== Statement ==<br />
Let <math>X</math> be a topological space, and <math>U, V\subset X</math> be open sets such that <math>X = U\cup V</math>, and <math>U</math>, <math>V</math> and <math>U\cap V</math> are path-connected. Take a basepoint <math>x_0\in U\cap V</math> with inclusion maps:<br />
<br />
<math display="block">i\colon U\cap V\hookrightarrow U,\quad j\colon U\cap V\hookrightarrow V,\quad k\colon U\hookrightarrow X,\quad l\colon V\hookrightarrow X,</math><br />
<br />
then the following diagram is a pushout:<br />
<div style="display: flex; justify-content: center; gap: 40px;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{quiver}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
{\pi_1(U\cap V,x_0)} & {\pi_1(U,x_0)} \\<br />
{\pi_1(V,x_0)} & {\pi_1(X,x_0)}<br />
\arrow["{i_*}", from=1-1, to=1-2]<br />
\arrow["{j_*}"', from=1-1, to=2-1]<br />
\arrow["{k_*}", from=1-2, to=2-2]<br />
\arrow["{l_*}"', from=2-1, to=2-2]<br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div></div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Commutator&diff=113Commutator2026-04-29T13:57:44Z<p>InfernalAtom683: Created page with "A '''commutator''' is an algebraic expression that measures the failure of two elements to commute. It occurs throughout abstract algebra, particularly in group theory, ring theory, and linear algebra. If two elements commute, their commutator is trivial. More generally, the commutator describes the obstruction to exchanging the order of two operations. Commutators are fundamental in the study of noncommutative structures and in the construction of i..."</p>
<hr />
<div>A '''commutator''' is an algebraic expression that measures the failure of two elements to [[commute]]. It occurs throughout [[abstract algebra]], particularly in [[group theory]], [[ring theory]], and [[linear algebra]].<br />
<br />
If two elements commute, their commutator is trivial. More generally, the commutator describes the obstruction to exchanging the order of two operations. Commutators are fundamental in the study of noncommutative structures and in the construction of invariants such as the [[derived subgroup]], the [[lower central series]], and the [[Lie bracket]].<br />
<br />
== Conventions ==<br />
<br />
In [[group theory]], two conventions are commonly used:<br />
<br />
* '''Left convention''':<br />
<math display="block"><br />
[a,b]=a^{-1}b^{-1}ab<br />
</math><br />
<br />
* '''Right convention''':<br />
<math display="block"><br />
[a,b]=aba^{-1}b^{-1}<br />
</math><br />
<br />
These differ by inversion:<br />
<math display="block"><br />
aba^{-1}b^{-1}=[b,a]^{-1}.<br />
</math><br />
<br />
Unless otherwise stated, this article uses the left convention.<br />
<br />
In [[ring theory]] and [[linear algebra]], the standard convention is<br />
<math display="block"><br />
[A,B]=AB-BA.<br />
</math><br />
<br />
== Definition ==<br />
<br />
=== Groups ===<br />
<br />
Let <math>G</math> be a [[group]], and let <math>a,b\in G</math>. Their commutator is<br />
<math display="block"><br />
[a,b]=a^{-1}b^{-1}ab.<br />
</math><br />
<br />
One has<br />
<math display="block"><br />
[a,b]=e \iff ab=ba,<br />
</math><br />
where <math>e</math> is the [[identity element]].<br />
<br />
=== Rings ===<br />
<br />
Let <math>R</math> be a [[ring]], and let <math>A,B\in R</math>. Their commutator is<br />
<math display="block"><br />
[A,B]=AB-BA.<br />
</math><br />
<br />
This vanishes precisely when <math>A</math> and <math>B</math> commute.<br />
<br />
=== Linear transformations ===<br />
<br />
For [[linear transformation]]s <math>A,B:V\to V</math> on a [[vector space]] <math>V</math>, equivalently for [[square matrix|square matrices]], the commutator is<br />
<math display="block"><br />
[A,B]=AB-BA.<br />
</math><br />
<br />
This is the ring commutator in the [[endomorphism ring]] <math>\operatorname{End}(V)</math>.<br />
<br />
== Properties ==<br />
<br />
=== Group identities ===<br />
<br />
For all <math>a,b,c\in G</math>,<br />
<math display="block"><br />
[a,b]^{-1}=[b,a],<br />
</math><br />
<br />
<math display="block"><br />
[a,a]=e,<br />
</math><br />
<br />
<math display="block"><br />
[a,b]^c=[a^c,b^c],<br />
</math><br />
where<br />
<math display="block"><br />
x^y=y^{-1}xy.<br />
</math><br />
<br />
Also,<br />
<math display="block"><br />
[ab,c]=[a,c]^b[b,c],<br />
</math><br />
<br />
<math display="block"><br />
[a,bc]=[a,c][a,b]^c.<br />
</math><br />
<br />
=== Ring identities ===<br />
<br />
For all <math>A,B,C</math>,<br />
<math display="block"><br />
[A,B]=-[B,A],<br />
</math><br />
<br />
<math display="block"><br />
[A+B,C]=[A,C]+[B,C],<br />
</math><br />
<br />
<math display="block"><br />
[A,BC]=[A,B]C+B[A,C].<br />
</math><br />
<br />
The commutator also satisfies the [[Jacobi identity]]:<br />
<math display="block"><br />
[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0.<br />
</math><br />
<br />
For matrices,<br />
<math display="block"><br />
\operatorname{tr}([A,B])=0.<br />
</math><br />
<br />
== Derived subgroup ==<br />
<br />
The subgroup generated by all group commutators is the '''[[derived subgroup]]''' or '''commutator subgroup''' of <math>G</math>:<br />
<math display="block"><br />
G'=[G,G]=\langle [a,b]\mid a,b\in G\rangle.<br />
</math><br />
<br />
It satisfies:<br />
<br />
* <math>G'</math> is a [[normal subgroup]]<br />
* <math>G/G'</math> is [[abelian group|abelian]]<br />
* <math>G</math> is abelian if and only if <math>G'=\{e\}</math><br />
<br />
The quotient<br />
<math display="block"><br />
G^{\mathrm{ab}}=G/G'<br />
</math><br />
is called the [[abelianization]] of <math>G</math>.<br />
<br />
== Lie algebras ==<br />
<br />
In a [[Lie algebra]], the bracket operation often arises from the commutator in an [[associative algebra]]:<br />
<math display="block"><br />
[x,y]=xy-yx.<br />
</math><br />
<br />
Thus every associative algebra determines a Lie algebra by using the commutator as its bracket.<br />
<br />
For the matrix algebra <math>M_n(F)</math>, this gives the Lie algebra<br />
<math display="block"><br />
\mathfrak{gl}(n,F).<br />
</math><br />
<br />
== Examples ==<br />
<br />
=== Symmetric group ===<br />
<br />
In the [[symmetric group]] <math>S_3</math>, let<br />
<math display="block"><br />
a=(1\;2),\qquad b=(2\;3).<br />
</math><br />
<br />
Then<br />
<math display="block"><br />
[a,b]=(1\;3\;2),<br />
</math><br />
so <math>a</math> and <math>b</math> do not commute.<br />
<br />
=== Matrices ===<br />
<br />
Let<br />
<math display="block"><br />
A=<br />
\begin{pmatrix}<br />
0&1\\<br />
0&0<br />
\end{pmatrix},<br />
\qquad<br />
B=<br />
\begin{pmatrix}<br />
0&0\\<br />
1&0<br />
\end{pmatrix}.<br />
</math><br />
<br />
Then<br />
<math display="block"><br />
[A,B]=<br />
\begin{pmatrix}<br />
1&0\\<br />
0&-1<br />
\end{pmatrix}.<br />
</math><br />
<br />
Hence <math>A</math> and <math>B</math> do not commute.<br />
<br />
== See also ==<br />
<br />
* [[Derived subgroup]]<br />
* [[Lower central series]]<br />
* [[Lie algebra]]<br />
* [[Jacobi identity]]<br />
* [[Abelianization]]<br />
* [[Noncommutative ring]]</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Compact_space&diff=111Compact space2026-04-14T08:11:00Z<p>InfernalAtom683: </p>
<hr />
<div>A '''compact''' [[topological space]] is one that behaves, in many respects, like a finite space, even if it is infinite. Specifically, a compact space is a topological space whose every open cover admits a finite subcover. Compactness is one of the most fundamental [[Topological property|topological properties]] in [[analysis]] and [[topology]].<br />
<br />
Intuitively, compactness can be understood as a generalization of being "[[Closed set|closed]] and [[Boundedness|bounded]]". In [[Euclidean space|Euclidean spaces]] <math>\mathbb{R}^n</math>, by the [[Heine–Borel theorem]], a set is compact if and only if it is closed and bounded.<br />
<br />
However, in a general topological space, a [[metric]] is typically not available, thus "boundedness" cannot be defined in a meaningful way. Therefore, an adopted definition is the one using open cover. In <math>\mathbb{R}^n</math>, this condition is equivalent to being closed and bounded, while still making sense in arbitrary topological spaces and preserving the essential properties of compact sets.<br />
<br />
== Definition ==<br />
A topological space <math>X</math> is compact if for every collection <math>\{U_i\}_{i\in I}</math> of opensets in <math>X</math> such that<br />
<math display="block">X=\bigcup_{i\in I}U_i,</math><br />
there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,U_{i_n}\}</math> such that<br />
<math display="block">X=\bigcup_{k=1}^n U_{i_k}.</math></div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homeomorphism&diff=110Homeomorphism2026-04-13T05:59:56Z<p>InfernalAtom683: </p>
<hr />
<div>[[File:Topology joke.jpg|thumb|250x250px|A homeomorphism that turns a coffee mug into a donut continuously.]]<br />
A '''homeomorphism''' is a special type of [[function]] between two [[Topological space|topological spaces]], that establishes that the two spaces are fundamentally the same from a topological perspective. Specifically, it is a [[Continuous function|continuous]] [[bijective]] function whose [[inverse function]] is also continuous. Homeomorphisms are the [[Isomorphism|isomorphisms]] in the [[category of topological spaces]] <math>\mathsf{Top}</math>, which preserves all [[topological properties]] of a topological space. If such a function exists between two spaces, they are said to be '''homeomorphic'''. <br />
<br />
Intuitively, two spaces are homeomorphic if one can be continuously deformed into the other by stretching, bending, and twisting, without cutting, tearing, or gluing. A typical intuitive example is that a mug with a handle is homeomorphic to a donut. This concept is distinct from [[Homotopy#Homotopy equivalence|homotopy equivalence]], which allows deformations that involve collapsing. For instance, a solid ball can be continuously shrunk to a point by a homotopy, but such a deformation is not a homeomorphism because it is not bijective and the inverse would not be continuous.<br />
<br />
== Definitions ==<br />
A function <math>f</math> between topological spaces <math>X</math> and <math>Y</math> is called a '''homeomorphism''', if:<br />
<br />
* <math>f</math> is continuous,<br />
* <math>f</math> is bijective,<br />
* <math>f^{-1}</math> is continuous.<br />
<br />
Two topological spaces <math>X</math> and <math>Y</math> are called '''homeomorphic''' if there exists a homeomorphism between them, denoted <math>X\cong Y</math>.<br />
<br />
=== Equivalent Definitions ===<br />
A homeomorphism is a bijection that is continuous and [[Open function|open]], or continuous and [[Closed function|closed]].<br />
<br />
== Properties ==<br />
{{Property|property=The composition of two homeomorphisms is again a homeomorphism.}}<br />
{{Proof|proof=Let <math>f\colon X \to Y</math> and <math>g\colon Y \to Z</math> be homeomorphisms. Then:<br />
<br />
* <math>g \circ f\colon X \to Z</math> is bijective, since the composition of two bijections is a bijection.<br />
<br />
* <math>g \circ f</math> is continuous, as the composition of two continuous functions.<br />
<br />
* The inverse is <math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}</math>, which is continuous because it is the composition of the continuous functions <math>g^{-1}</math> and <math>f^{-1}</math>.<br />
<br />
Thus <math>g \circ f</math> satisfies all requirements of a homeomorphism.}}<br />
{{Proof|title=Proof via universal property<br />
|proof=The following commutative diagrams exhibit <math>f</math> as an isomorphism:<br />
<div style="display: flex; justify-content: center; gap: 40px;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
X \arrow[r,"f"] \arrow[rd,"\mathrm{id}_X"'] & Y \arrow[d,"f^{-1}"]\\ <br />
& X <br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
Y \arrow[r,"f^{-1}"] \arrow[rd,"\mathrm{id}_Y"'] & X \arrow[d,"f"]\\ <br />
& Y <br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
And the following commmutative diagrams are for <math>g</math>:<br />
<div style="display: flex; justify-content: center; gap: 40px;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
Y \arrow[r,"g"] \arrow[rd,"\mathrm{id}_Y"'] & Z \arrow[d,"g^{-1}"]\\ <br />
& Y <br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
Z \arrow[r,"g^{-1}"] \arrow[rd,"\mathrm{id}_Z"'] & Y \arrow[d,"g"]\\ <br />
& Z <br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
Note that the commutative triangles for <math>f</math> and <math>g</math> paste to yeild the commutative triangle for <math>g\circ f</math>:<br />
<div style="display: flex; justify-content: center; gap: 40px;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.3] {<br />
\begin{tikzcd}<br />
& Y && \\<br />
X && Z \\<br />
&&& Y \\<br />
&& X<br />
\arrow["g", from=1-2, to=2-3]<br />
\arrow["f", from=2-1, to=1-2]<br />
\arrow["{g\circ f}", from=2-1, to=2-3]<br />
\arrow["{\mathrm{id}_X}"', from=2-1, to=4-3]<br />
\arrow["{g^{-1}}", from=2-3, to=3-4]<br />
\arrow["{f^{-1}\circ g^{-1}}", from=2-3, to=4-3]<br />
\arrow["{f^{-1}}", from=3-4, to=4-3]<br />
\end{tikzcd}};<br />
\end{tikzpicture} <br />
\end{document}<br />
</kroki><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.3] {<br />
\begin{tikzcd}<br />
& Y && \\<br />
Z && X \\<br />
&&& Y \\<br />
&& Z<br />
\arrow["{f^{-1}}", from=1-2, to=2-3]<br />
\arrow["{g^{-1}}", from=2-1, to=1-2]<br />
\arrow["{(g\circ f)^{-1}}", from=2-1, to=2-3]<br />
\arrow["{\mathrm{id}_Z}"', from=2-1, to=4-3]<br />
\arrow["f", from=2-3, to=3-4]<br />
\arrow["{g\circ f}", from=2-3, to=4-3]<br />
\arrow["g", from=3-4, to=4-3]<br />
\end{tikzcd}};<br />
\end{tikzpicture} <br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
}}<br />
<br />
<br />
{{Property|property=The inverse of a homeomorphism is again a homeomorphism.}}<br />
{{Proof|proof=Let <math>f\colon X \to Y</math> be a homeomorphism. Then:<br />
* <math>f^{-1}</math> is continuous by definition,<br />
* <math>f^{-1}</math> is bijective, since the inverse of a bijection is again a bijection,<br />
* <math>\left(f^{-1}\right)^{-1}=f</math> is continuous by definition.}}<br />
<br />
<br />
{{Property|property=Homeomorphism is an [[equivalence relation]].}}<br />
<br />
{{Proof|proof=<br />
* '''Reflexivity''': The identity map <math>\operatorname{id}_X\colon X\to X</math> is a continuous bijection on any topological space <math>X</math>, whose inverse is itself. Thus <math>\operatorname{id}_X</math> is a homeomorphism.<br />
* '''Symmetry''': If <math>f\colon X\to Y</math> is a homeomorphism, then its inverse <math>f^{-1}\colon Y\to X</math> is again a homeomorphism.<br />
* '''Transitivity''': If <math>f\colon X\to Y</math> and <math>g\colon Y\to Z</math> are homeomorphisms, then <math>g\circ f: X\to Z</math> is again a homeomorphism.<br />
}}<br />
<br />
== Examples ==<br />
<br />
=== Open interval ===<br />
The [[open interval]] <math>(0,1)</math> is homeomorphic to <math>\mathbb{R}</math>.<br />
{{Proof|proof=The map <math>f\colon (0,1)\to \mathbb{R}</math> defined by<br />
<math display="block">f(x)=\tan\left(\pi\left(x-\dfrac12\right)\right)</math><br />
is a homeomorphism. Indeed, <math>f</math> is continuous because it is a composition of continuous functions. The restriction <math>\tan\colon (-\pi/2,\pi/2)\to\mathbb{R}</math> is bijective with continuous inverse <math>\arctan\colon \mathbb{R}\to(-\pi/2,\pi/2)</math>. Therefore <math>f</math> is bijective and its inverse<br />
<math display="block">f^{-1}(y)=\dfrac1\pi\arctan(y)+\dfrac12</math><br />
is continuous. Thus <math>f</math> is a homeomorphism.}}<br />
<br />
=== Stereographic projection ===<br />
The [[Euclidean plane]] <math>\mathbb{R}^2</math> is homeomorphic to the [[2-sphere]] minus one point, denoted <math>S^2 \setminus \{N\}</math> where <math>N=(0,0,1)</math> is the [[north pole]].<br />
<br />
{{Proof|proof=<br />
<div style="float:right; width:320px; margin:0 0 0.5em 1em;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=15pt]{standalone}<br />
\usepackage{tikz-3dplot}<br />
\usetikzlibrary{calc, arrows.meta}<br />
\begin{document}<br />
\def\viewTheta{70}<br />
\def\viewPhi{20}<br />
\tdplotsetmaincoords{\viewTheta}{\viewPhi}<br />
<br />
\begin{tikzpicture}[tdplot_main_coords, scale=2, line cap=round, line join=round]<br />
<br />
\def\R{1}<br />
\coordinate (O) at (0,0,0);<br />
\coordinate (N) at (0,0,\R);<br />
<br />
\def\thetaS{60}<br />
\def\phiS{30}<br />
\pgfmathsetmacro{\px}{\R * sin(\thetaS) * cos(\phiS)}<br />
\pgfmathsetmacro{\py}{\R * sin(\thetaS) * sin(\phiS)}<br />
\pgfmathsetmacro{\pz}{\R * cos(\thetaS)}<br />
\coordinate (P) at (\px, \py, \pz);<br />
<br />
\pgfmathsetmacro{\ux}{\px / (1 - \pz)}<br />
\pgfmathsetmacro{\uy}{\py / (1 - \pz)}<br />
\coordinate (Pprime) at (\ux, \uy, 0);<br />
<br />
\pgfmathsetmacro{\eqFrontStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\eqBackStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqBackEnd}{\viewPhi+180}<br />
<br />
\pgfmathsetmacro{\cotViewTheta}{cos(\viewTheta)/sin(\viewTheta)}<br />
\pgfmathsetmacro{\cotThetaS}{cos(\thetaS)/sin(\thetaS)}<br />
\pgfmathsetmacro{\cosAlpha}{max(min(-\cotThetaS * \cotViewTheta, 1), -1)}<br />
\pgfmathsetmacro{\alpha}{acos(\cosAlpha)}<br />
<br />
\pgfmathsetmacro{\latFrontStart}{\viewPhi-180}<br />
\pgfmathsetmacro{\latFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\latBackStart}{\viewPhi}<br />
\pgfmathsetmacro{\latBackEnd}{\viewPhi+180}<br />
<br />
\draw[thick, black] (-1.2,0,0) -- (0,0,0);<br />
\draw[thick, black] (0,-3,0) -- (0,0,0);<br />
\draw[thick, black] (0,0,-1.2) -- (0,0,0);<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (2.2,0,0) node[anchor=north east]{$x$};<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (0,3.0,0) node[anchor=north west]{$y$};<br />
\draw[thick, dashed] (0,0,0) -- (N);<br />
\draw[thick, ->, >=Stealth] (N) -- (0,0,1.8) node[anchor=south]{$z$};<br />
\begin{scope}[tdplot_screen_coords]<br />
\shade[ball color=cyan, opacity=0.15] (0,0) circle (\R);<br />
\draw[cyan!60!blue, thick] (0,0) circle (\R);<br />
\end{scope}<br />
<br />
\tdplotdrawarc[cyan!60!blue, thick]{(O)}{\R}{\eqFrontStart}{\eqFrontEnd}{}{}<br />
\tdplotdrawarc[cyan!60!blue, thin, dashed]{(O)}{\R}{\eqBackStart}{\eqBackEnd}{}{}<br />
<br />
\pgfmathsetmacro{\rLat}{\R * sin(\thetaS)}<br />
\coordinate (CenterLat) at (0,0,\pz);<br />
<br />
\tdplotsetrotatedcoords{0}{0}{0}<br />
\tdplotsetrotatedcoordsorigin{(CenterLat)}<br />
\tdplotdrawarc[cyan!70!black, thin, dashed, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latBackStart}{\latBackEnd}{}{}<br />
\tdplotdrawarc[cyan!70!black, thin, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latFrontStart}{\latFrontEnd}{}{}<br />
\tdplotsetrotatedcoordsorigin{(O)}<br />
<br />
\draw[red, thick, dashed] (N) -- (P);<br />
\draw[red, thick, ->, >=Stealth] (P) -- (Pprime);<br />
<br />
\fill[black] (N) circle (0.8pt) node[anchor=south east] {$N$};<br />
\fill[red] (P) circle (1pt) node[anchor=south west, text=black] {$(x,y,z)$};<br />
\fill[red] (Pprime) circle (1pt) node[anchor=north west, text=black] {$p(x,y,z) = (u,v)$};<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
Define the [[stereographic projection]] <math>p\colon S^2 \setminus \{N\} \to \mathbb{R}^2</math> by<br />
<math display="block">p(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right).</math><br />
This map is continuous because it is a rational function with denominator nonzero (since <math>z<1</math> on <math>S^2\setminus\{N\}</math>).<br />
<br />
The inverse map <math>p^{-1}\colon \mathbb{R}^2 \to S^2 \setminus \{N\}</math> is given by<br />
<math display="block">p^{-1}(u,v) = \left( \frac{2u}{u^2+v^2+1}, \frac{2v}{u^2+v^2+1}, \frac{u^2+v^2-1}{u^2+v^2+1} \right).</math><br />
This is also continuous as a composition of continuous functions. One verifies that <math>p \circ p^{-1} = \text{id}_{\mathbb{R}^2}</math> and <math>p^{-1} \circ p = \operatorname{id}_{S^2\setminus\{N\}}</math> by direct substitution. Hence <math>p</math> is a homeomorphism.<br />
}}<br />
<br />
=== Quotient space ===<br />
The unit interval <math>[0,1]</math> with the endpoints identified (the quotient space <math>[0,1]/\sim</math> where <math>0\sim 1</math>) is homeomorphic to the circle <math>S^1</math>.<br />
<br />
{{Proof|proof=Define the map <math>f\colon [0,1] \to S^1</math> by <math display="block">f(t)=(\cos(2\pi t), \sin(2\pi t)).</math> This map is continuous and [[Surjection|surjective]], and satisfies <math>f(0)=f(1)=(1,0)</math>.<br />
<br />
Consider the equivalence relation <math>\sim</math>, and let <math>q\colon [0,1]\to [0,1]/\sim</math> be the [[quotient map]]. By the [[universal property]] of the quotient map, there exists a unique continuous map <math>\tilde{f}\colon [0,1]/\sim \to S^1</math> such that <math>\tilde{f} \circ q = f</math>; that is, the following diagram commutes:<br />
<div style="text-align: center;"><br />
<br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
{[0,1]} && {S^1} \\<br />
& {[0,1]/{\sim}} \arrow["f", from=1-1, to=1-3]<br />
\arrow["q"', from=1-1, to=2-2]<br />
\arrow["{\exists! \tilde{f}}"', dashed, from=2-2, to=1-3]<br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
The map <math>\tilde{f}</math> is bijective because:<br />
* Surjectivity follows from surjectivity of <math>f</math>;<br />
* [[Injection|Injectivity]] holds because <math display="block">\tilde{f}([t])=\tilde{f}([s])\Rightarrow t=s \text{ or } \{t,s\}=\{0,1\},</math> but in the latter case <math>[t]=[s]</math> in the quotient. <br />
<br />
Hence <math>\tilde{f}</math> is a continuous bijection.<br />
<br />
The space <math>[0,1]/\sim</math> is compact as the quotient of a [[compact space]], and <math>S^1</math> is [[Hausdorff space|Hausdorff]]. By the [[Compact-to-Hausdorff theorem]], a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.<br />
<br />
Therefore <math>\tilde{f}</math> is a homeomorphism.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz">\documentclass[tikz,border=10pt]{standalone}<br />
\usepackage{amsmath}<br />
\usetikzlibrary{arrows.meta,calc}<br />
\begin{document}<br />
<br />
\begin{tikzpicture}[<br />
>={Stealth[scale=1.1]},<br />
dot/.style={circle,fill=black,inner sep=1.6pt},<br />
label text/.style={font=\Large,align=center}<br />
]<br />
<br />
\def\r{1.4}<br />
\def\gap{50}<br />
<br />
\coordinate (C1) at (0,0);<br />
\coordinate (C2) at (5.5,0);<br />
\coordinate (C3) at (11,0);<br />
<br />
\begin{scope}[shift={(C1)}]<br />
\draw[thick] (-\r,0) coordinate (A) -- (\r,0) coordinate (B);<br />
\node[dot] at (A) {};<br />
\node[dot] at (B) {};<br />
\node[label text] at (0,-2.6) {$[0,1]$};<br />
\coordinate (R1) at (\r,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C1)+(2,0)$) -- ($(C2)+(-2,0)$)<br />
node[midway,above] {$q$};<br />
<br />
\begin{scope}[shift={(C2)}]<br />
\draw[thick]<br />
(180-\gap:\r)<br />
arc[start angle=180-\gap,end angle=360+\gap,radius=\r];<br />
<br />
\node[dot] (L) at (180-\gap:\r) {};<br />
\node[dot] (R) at (\gap:\r) {};<br />
<br />
\draw[dashed,->,thick]<br />
(L) .. controls +(0,0) and +(-0.8,-0.1) .. (90:\r);<br />
\draw[dashed,->,thick]<br />
(R) .. controls +(0,0) and +(0.8,-0.1) .. (90:\r);<br />
<br />
\node[label text] at (0,-2.6)<br />
{$[0,1]/\sim$ \\[-0.4ex]\normalsize $(0\sim1)$};<br />
<br />
\coordinate (R2) at (2,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C2)+(2,0)$) -- ($(C3)+(-2,0)$)<br />
node[midway,above] {$\overset{\tilde{f}}{\cong}$};<br />
<br />
\begin{scope}[shift={(C3)}]<br />
\draw[thick] (0,0) circle (\r);<br />
\node[dot] at (90:\r) {};<br />
\node[label text] at (0,-2.6) {$S^{1}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}</kroki><br />
</div><br />
<br />
}}<br />
<br />
== Counterexamples ==<br />
<math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>.<br />
<br />
{{Proof|proof=For contradiction, suppose that there exists a homeomorphism <math>f\colon \mathbb{R}\to\mathbb{R}^2</math>.<br />
<br />
Consider the subspace <math>\mathbb{R}\setminus\{0\}</math> of <math>\mathbb{R}</math>. The [[restriction]] on it, <math>\left.f\right|_{\mathbb{R}\setminus\{0\}}\colon \mathbb{R}\setminus\{0\}\to \mathbb{R}^2\setminus\{f(0)\}</math> is also a homeomorphism.<br />
<br />
However, <math>\mathbb{R}\setminus\{0\}</math> has two connected components, <math>(-\infty,0)</math> and <math>(0,\infty)</math>, while <math>\mathbb{R}^2\setminus\{f(0)\}</math> is connected, which contradicts the assumption that the two spaces are homeomorphic.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz, border=10pt]{standalone}<br />
\usepackage{amsmath, amssymb}<br />
<br />
\begin{document}<br />
\begin{tikzpicture}<br />
\begin{scope}[xshift=-5cm]<br />
\draw[thick] (-3, 0) -- (3, 0);<br />
<br />
\filldraw[fill=white, draw=black, thick] (0, 0) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2.5, 0) -- (-1, 0);<br />
\fill[cyan!60!blue] (-2.5, 0) circle (2pt);<br />
\fill[cyan!60!blue] (-1, 0) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (1.3, 0) -- (2.2, 0);<br />
\fill[cyan!60!blue] (1.3, 0) circle (2pt);<br />
\fill[cyan!60!blue] (2.2, 0) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -2) {$\mathbb{R} \setminus \{0\}$};<br />
\end{scope}<br />
<br />
\begin{scope}[xshift=4cm]<br />
\draw[thick] (-3.5, -2.5) -- (3.5, -2.5) -- (3.5, 2.5) -- (-3.5, 2.5) -- cycle;<br />
<br />
\filldraw[fill=white, draw=black, thick] (-0.3, -0.2) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2, 1.5) .. controls (-0.5, 1) and (-0.8, -1) .. (-1.2, -1.8);<br />
\fill[cyan!60!blue] (-2, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (-1.2, -1.8) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 1.5) .. controls (1.5, 1.8) and (2.5, 1) .. (2, 0.5);<br />
\fill[cyan!60!blue] (-0.3, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (2, 0.5) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 0.5) .. controls (1, 0) and (1, -1) .. (2.5, -2);<br />
\fill[cyan!60!blue] (-0.3, 0.5) circle (2pt);<br />
\fill[cyan!60!blue] (2.5, -2) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.5, -0.8) .. controls (0, -1.8) and (1, -1.5) .. (0.8, -1);<br />
\fill[cyan!60!blue] (-0.5, -0.8) circle (2pt);<br />
\fill[cyan!60!blue] (0.8, -1) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -3.5) {$\mathbb{R}^2 \setminus \{f(0)\}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
Hence, no such homeomorphism exists; therefore <math>\mathbb{R}</math> is not homeomorphic to <math>\mathbb{R}^2</math>}}<br />
<br />
The map from the interval <math>[0,1)</math> to the 1-sphere <math>S^1</math>,<br />
<math display="block">\phi\colon [0,1)\to S^1,\quad x\mapsto e^{2\pi ix}</math><br />
is continuous and bijective, but not a homeomorphism. <br />
{{Proof|proof=<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass{article}<br />
\usepackage{tikz}<br />
\usetikzlibrary{arrows.meta}<br />
<br />
\begin{document}<br />
<br />
\begin{tikzpicture}<br />
\draw (0,0) -- (4,0);<br />
<br />
\draw (0.15, 0.25) -- (0, 0.25) -- (0, -0.25) -- (0.15, -0.25);<br />
<br />
\draw (3.9, 0.25) to[bend left=45] (3.9, -0.25);<br />
<br />
\draw[-{Stealth[length=3mm, width=2mm]}] (4.5, 1.2) to[bend left=30] node[above=2pt] {$\phi$} (7.0, 1.2);<br />
<br />
\draw (9.5, 0) circle (2);<br />
<br />
\begin{scope}[rotate around={90:(11.5,0)}]<br />
\draw (11.39, 0.25) to[bend left=45] (11.39, -0.25);<br />
\draw (11.5, -0.25) -- (11.5, 0.25);<br />
\draw (11.5, 0.25) -- (11.65, 0.25);<br />
\draw (11.5, -0.25) -- (11.65, -0.25);<br />
<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
The map <math>\phi</math> is:<br />
* Continuous, as it is the composition of continuous maps <math>x\mapsto 2\pi x</math> and <math>t\mapsto e^{it}</math>.<br />
* Injective, because if <math>e^{2\pi i x_1}=e^{2\pi i x_2}</math>, then <math>x_1-x_2\in \mathbb{Z}</math>. Since <math>x_1,x_2\in [0,1)</math>, it follows that <math>x_1=x_2</math>.<br />
* Surjective, since every point of <math>S^1</math> can be written as <math>e^{2\pi i x}</math> for some <math>x\in [0,1)</math>.<br />
Hence <math>\phi</math> is a continuous bijection.<br />
<br />
However, <math>\phi</math> is not a homeomorphism.<br />
<br />
Consider the sequence<math display="block">z_n = e^{2\pi i (1-\tfrac{1}{n})} \in S^1.</math><br />
Then <math>z_n \to 1 = e^{2\pi i \cdot 0}</math> in <math>S^1</math>. But<br />
<math display="block">\phi^{-1}(z_n) = 1-\frac{1}{n} \to 1,</math><br />
which does not converge to <math>\phi^{-1}(1)=0</math> in <math>[0,1)</math>. Thus <math>\phi^{-1}</math> is not continuous.<br />
}}<br />
<br />
== Topological invariants ==<br />
A [[topological invariant]] is a property of a topological space that is preserved under homeomorphisms. In other words, if two spaces are homeomorphic, they either both possess the property or both do not. Invariants are the important tools to classify topological spaces. If two spaces differ in any topological invariant, they cannot be homeomorphic. Conversely, showing that two spaces share many invariants is often the first step on proving they are homeomorpic, though it is never sufficient by itself.<br />
<br />
=== Common topological invariants ===<br />
<br />
* [[Connectedness]]<br />
* [[Compactness]]<br />
* [[Hausdorff space|Hausdorff property]]<br />
* [[Cardinality]] of the space<br />
<br />
=== Algebraic invariants ===<br />
More powerful invariants come from [[algebraic topology]], which assigns algebraic objects to topological spaces.<br />
<br />
* [[Fundamental group]]<br />
* [[Homology group]]<br />
* [[Higher homotopy group]]<br />
<br />
==Homeomorphism group==<br />
The collection of all [[autohomeomorphisms]] of a topological space <math>X</math> forms a [[group]] under composition operation, known as the homeomorphism group of <math>X</math>, denoted <math>\operatorname{Homeo}(X)</math>. The homeomorphism group captures the symmetry in topology. It describes the ways in which a topological space can be continuously transformed onto itself.<br />
<br />
The homeomorphism group <math>\operatorname{Homeo}(X)</math> is a faithful [[group action]] on its underlying set <math>X</math>. It moves points in <math>X</math> continuously onto <math>X</math> itself, and the topological structure of <math>X</math> is also reflected in the algebraic invariants such as the [[Orbit|orbits]] and [[Stabilizer|stabilizers]] of the action.<br />
<br />
For example, consider the 2-sphere <math>S^2</math> as a thin rubber membrane tightly wraped around a ball. Each autohomeomorphism of <math>S^2</math>, which is an element in <math>\operatorname{Homeo}(S^2)</math>, corresponds to a continuous deformation of this membrane. This operation can be stretching, bending, twisting, or any composition of these operations, so the rubber always remains attached to the ball.<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=5mm]{standalone}<br />
\usetikzlibrary{arrows.meta, positioning, calc}<br />
\usepackage{amssymb}<br />
\begin{document}<br />
\begin{tikzpicture}[scale=1.2, >=Stealth]<br />
\begin{scope}[shift={(0,0)}]<br />
\shade[ball color=cyan, opacity=0.2] (0,0) circle (2);<br />
<br />
\draw[cyan!60!blue, thick] (-2,0) arc (180:360:2 and 0.25);<br />
\draw[cyan!60!blue, thin, dashed] (2,0) arc (0:180:2 and 0.25);<br />
<br />
\foreach \y in {-1.2, -0.6, 0.6, 1.2} {<br />
\pgfmathsetmacro{\x}{sqrt(4-\y*\y)}<br />
\draw[cyan!70!black, thin] (-\x, \y) arc (180:360:\x cm and 0.25cm);<br />
\draw[cyan!70!black, thin, dashed] (\x, \y) arc (0:180:\x cm and 0.25cm);<br />
}<br />
<br />
\draw[cyan!70!black, thin] (0,2) arc (90:-90:0.6 and 2);<br />
\draw[cyan!70!black, thin] (0,2) arc (90:-90:1.3 and 2);<br />
\draw[cyan!70!black, thin] (0,2) arc (90:270:0.6 and 2);<br />
\draw[cyan!70!black, thin] (0,2) arc (90:270:1.3 and 2);<br />
\draw[cyan!70!black, thin] (0,2) -- (0,-2);<br />
<br />
\filldraw[fill=yellow!60, draw=yellow!60!black, thick, opacity=0.7]<br />
(0.7, 1.1) .. controls (1.0, 1.3) and (1.3, 1.1) .. (1.4, 0.5) <br />
.. controls (1.1, 0.1) and (0.1, 0.6) .. (0.7, 1.1) <br />
-- cycle;<br />
\node[yellow!60!black, font=\bfseries] at (1.1, 0.9) {$U$};<br />
<br />
\filldraw[black] (0.8, 0.9) circle (1pt) node[anchor=south east, scale=0.8] {$x$};<br />
\filldraw[black] (1.1, 0.7) circle (1pt) node[anchor=north west, scale=0.8] {$y$};<br />
<br />
\node[below, text=black] at (0,-2.3) {$X$};<br />
\draw[cyan!60!blue, thick] (0,0) circle (2);<br />
\end{scope}<br />
<br />
\draw[->, thick, black] (2.5, 0) -- (4.5, 0) <br />
node[midway, above, font=\small] {$\phi \in \mathrm{Homeo}(X)$};<br />
<br />
\begin{scope}[shift={(7,0)}]<br />
\shade[ball color=cyan, opacity=0.2] (0,0) circle (2);<br />
<br />
\draw[cyan!60!blue, thick] (-2.0,0.0) .. controls (-0.9, -0.3) and (0.9, 0.3) .. (2.0, 0.0);<br />
\draw[cyan!60!blue, thin, dashed] (2.0,0.0) .. controls (0.9, 0.6) and (-0.9, 0.0) .. (-2.0, 0.0);<br />
<br />
\draw[cyan!70!black, thin] (-1.6, 1.2) .. controls (-0.6, 0.9) and (0.6, 1.5) .. (1.6, 1.2);<br />
\draw[cyan!70!black, thin, dashed] (1.6, 1.2) .. controls (0.6, 1.6) and (-0.6, 1.0) .. (-1.6, 1.2);<br />
\draw[cyan!70!black, thin] (-1.9, 0.6) .. controls (-0.9, 0.3) and (0.9, 0.9) .. (1.9, 0.6);<br />
\draw[cyan!70!black, thin, dashed] (1.9, 0.6) .. controls (0.9, 1.1) and (-0.9, 0.5) .. (-1.9, 0.6);<br />
\draw[cyan!70!black, thin] (-1.9, -0.6) .. controls (-0.9, -0.9) and (0.9, -0.3) .. (1.9, -0.6);<br />
\draw[cyan!70!black, thin, dashed] (1.9, -0.6) .. controls (0.9, -0.1) and (-0.9, -0.5) .. (-1.9, -0.6);<br />
\draw[cyan!70!black, thin] (-1.6, -1.2) .. controls (-0.6, -1.5) and (0.6, -0.9) .. (1.6, -1.2);<br />
\draw[cyan!70!black, thin, dashed] (1.6, -1.2) .. controls (0.6, -0.8) and (-0.6, -1.4) .. (-1.6, -1.2);<br />
<br />
\draw[cyan!70!black, thin] (0, 2) .. controls (-1.9, 0.8) and (-1.7, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin] (0, 2) .. controls (-1.1, 0.8) and (-0.9, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin] (0, 2) .. controls (-0.3, 0.8) and (-0.1, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin] (0, 2) .. controls (0.5, 0.8) and (0.7, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin] (0, 2) .. controls (1.3, 0.8) and (1.5, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin, dashed] (0, 2) .. controls (-1.3, 0.8) and (-0.8, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin, dashed] (0, 2) .. controls (0.9, 0.8) and (0.5, -1.2) .. (0, -2);<br />
<br />
\filldraw[fill=yellow!60, draw=yellow!60!black, thick, opacity=0.7]<br />
(0.05, 1.3) <br />
.. controls (0.4, 1.4) and (0.9, 1.5) .. (1.2, 0.7) <br />
.. controls (0.9, 0.3) and (0.4, 0.15) .. (0.05, 0.5) <br />
.. controls (-0.1, 0.8) and (-0.1, 1.1) .. (0.05, 1.3) <br />
-- cycle;<br />
\node[yellow!60!black, font=\bfseries] at (0.6, 0.95) {$\phi(U)$};<br />
<br />
\filldraw[black] (0.4, 1.1) circle (1pt) node[anchor=south east, scale=0.8] {$\phi(x)$};<br />
\filldraw[black] (0.8, 0.6) circle (1pt) node[anchor=north west, scale=0.8] {$\phi(y)$};<br />
<br />
\node[below, text=black] at (0,-2.3) {$\phi\cdot X$};<br />
\draw[cyan!60!blue, thick] (0,0) circle (2);<br />
\end{scope}<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
Under the natural action of <math>\operatorname{Homeo}(S^2)</math>, every point on the sphere can be moved continuously to any other point. This example shows how the homeomorphism group captures the symmetry of a topological space in the perspective of continuity.<br />
<br />
==See also==<br />
* [[Homotopy]]<br />
* [[Topology]]<br />
* [[Homeomorphism group]]<br />
<br />
==Terminology==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homeomorphism | homéomorphisme | Homeomorphismus | 同胚 | 同胚 | 同相写像 }}<br />
{{Terminology_table/row | homeomorphic | homéomorphe | homeomorph | 同胚的 | 同胚的 | 同相 }}<br />
{{Terminology_table/row | topological invariant | invariant topologique | topologische Invariante | 拓扑不变量 | 拓撲不變量 | 位相不変量 }}<br />
{{Terminology_table/row | autohomeomorphism | autohoméomorphisme | Selbsthomöomorphismus | 自同胚 | 自同胚 | 自己同相写像 }}<br />
{{Terminology_table/row | homeomorphism group | groupe des homéomorphismes | Homöomorphismengruppe | 同胚群 | 同胚群 | 同相群 }}<br />
}}<br />
<br />
[[Category:Topology]]</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homeomorphism&diff=109Homeomorphism2026-04-12T12:27:58Z<p>InfernalAtom683: </p>
<hr />
<div>[[File:Topology joke.jpg|thumb|250x250px|A homeomorphism that turns a coffee mug into a donut continuously.]]<br />
A '''homeomorphism''' is a special type of [[function]] between two [[Topological space|topological spaces]], that establishes that the two spaces are fundamentally the same from a topological perspective. Specifically, it is a [[Continuous function|continuous]] [[bijective]] function whose [[inverse function]] is also continuous. Homeomorphisms are the [[Isomorphism|isomorphisms]] in the [[category of topological spaces]] <math>\mathsf{Top}</math>, which preserves all [[topological properties]] of a topological space. If such a function exists between two spaces, they are said to be '''homeomorphic'''. <br />
<br />
Intuitively, two spaces are homeomorphic if one can be continuously deformed into the other by stretching, bending, and twisting, without cutting, tearing, or gluing. A typical intuitive example is that a mug with a handle is homeomorphic to a donut. This concept is distinct from [[Homotopy#Homotopy equivalence|homotopy equivalence]], which allows deformations that involve collapsing. For instance, a solid ball can be continuously shrunk to a point by a homotopy, but such a deformation is not a homeomorphism because it is not bijective and the inverse would not be continuous.<br />
<br />
== Definitions ==<br />
A function <math>f</math> between topological spaces <math>X</math> and <math>Y</math> is called a '''homeomorphism''', if:<br />
<br />
* <math>f</math> is continuous,<br />
* <math>f</math> is bijective,<br />
* <math>f^{-1}</math> is continuous.<br />
<br />
Two topological spaces <math>X</math> and <math>Y</math> are called '''homeomorphic''' if there exists a homeomorphism between them, denoted <math>X\cong Y</math>.<br />
<br />
=== Equivalent Definitions ===<br />
A homeomorphism is a bijection that is continuous and [[Open function|open]], or continuous and [[Closed function|closed]].<br />
<br />
== Properties ==<br />
{{Property|property=The composition of two homeomorphisms is again a homeomorphism.}}<br />
{{Proof|proof=Let <math>f\colon X \to Y</math> and <math>g\colon Y \to Z</math> be homeomorphisms. Then:<br />
<br />
* <math>g \circ f\colon X \to Z</math> is bijective, since the composition of two bijections is a bijection.<br />
<br />
* <math>g \circ f</math> is continuous, as the composition of two continuous functions.<br />
<br />
* The inverse is <math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}</math>, which is continuous because it is the composition of the continuous functions <math>g^{-1}</math> and <math>f^{-1}</math>.<br />
<br />
Thus <math>g \circ f</math> satisfies all requirements of a homeomorphism.}}<br />
{{Proof|title=Proof via universal property<br />
|proof=The following commutative diagrams exhibit <math>f</math> as an isomorphism:<br />
<div style="display: flex; justify-content: center; gap: 40px;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
X \arrow[r,"f"] \arrow[rd,"\mathrm{id}_X"'] & Y \arrow[d,"f^{-1}"]\\ <br />
& X <br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
Y \arrow[r,"f^{-1}"] \arrow[rd,"\mathrm{id}_Y"'] & X \arrow[d,"f"]\\ <br />
& Y <br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
And the following commmutative diagrams are for <math>g</math>:<br />
<div style="display: flex; justify-content: center; gap: 40px;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
Y \arrow[r,"g"] \arrow[rd,"\mathrm{id}_Y"'] & Z \arrow[d,"g^{-1}"]\\ <br />
& Y <br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
Z \arrow[r,"g^{-1}"] \arrow[rd,"\mathrm{id}_Z"'] & Y \arrow[d,"g"]\\ <br />
& Z <br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
Note that the commutative triangles for <math>f</math> and <math>g</math> paste to yeild the commutative triangle for <math>g\circ f</math>:<br />
<div style="display: flex; justify-content: center; gap: 40px;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
& Y && \\<br />
X && Z \\<br />
&&& Y \\<br />
&& X<br />
\arrow["g", from=1-2, to=2-3]<br />
\arrow["f", from=2-1, to=1-2]<br />
\arrow["{g\circ f}", from=2-1, to=2-3]<br />
\arrow["{\mathrm{id}_X}"', from=2-1, to=4-3]<br />
\arrow["{g^{-1}}", from=2-3, to=3-4]<br />
\arrow["{f^{-1}\circ g^{-1}}", from=2-3, to=4-3]<br />
\arrow["{f^{-1}}", from=3-4, to=4-3]<br />
\end{tikzcd}};<br />
\end{tikzpicture} <br />
\end{document}<br />
</kroki><br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
& Y && \\<br />
Z && X \\<br />
&&& Y \\<br />
&& Z<br />
\arrow["{f^{-1}}", from=1-2, to=2-3]<br />
\arrow["{g^{-1}}", from=2-1, to=1-2]<br />
\arrow["{(g\circ f)^{-1}}", from=2-1, to=2-3]<br />
\arrow["{\mathrm{id}_Z}"', from=2-1, to=4-3]<br />
\arrow["f", from=2-3, to=3-4]<br />
\arrow["{g\circ f}", from=2-3, to=4-3]<br />
\arrow["g", from=3-4, to=4-3]<br />
\end{tikzcd}};<br />
\end{tikzpicture} <br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
}}<br />
<br />
<br />
{{Property|property=The inverse of a homeomorphism is again a homeomorphism.}}<br />
{{Proof|proof=Let <math>f\colon X \to Y</math> be a homeomorphism. Then:<br />
* <math>f^{-1}</math> is continuous by definition,<br />
* <math>f^{-1}</math> is bijective, since the inverse of a bijection is again a bijection,<br />
* <math>\left(f^{-1}\right)^{-1}=f</math> is continuous by definition.}}<br />
<br />
<br />
{{Property|property=Homeomorphism is an [[equivalence relation]].}}<br />
<br />
{{Proof|proof=<br />
* '''Reflexivity''': The identity map <math>\operatorname{id}_X\colon X\to X</math> is a continuous bijection on any topological space <math>X</math>, whose inverse is itself. Thus <math>\operatorname{id}_X</math> is a homeomorphism.<br />
* '''Symmetry''': If <math>f\colon X\to Y</math> is a homeomorphism, then its inverse <math>f^{-1}\colon Y\to X</math> is again a homeomorphism.<br />
* '''Transitivity''': If <math>f\colon X\to Y</math> and <math>g\colon Y\to Z</math> are homeomorphisms, then <math>g\circ f: X\to Z</math> is again a homeomorphism.<br />
}}<br />
<br />
== Examples ==<br />
<br />
=== Open interval ===<br />
The [[open interval]] <math>(0,1)</math> is homeomorphic to <math>\mathbb{R}</math>.<br />
{{Proof|proof=The map <math>f\colon (0,1)\to \mathbb{R}</math> defined by<br />
<math display="block">f(x)=\tan\left(\pi\left(x-\dfrac12\right)\right)</math><br />
is a homeomorphism. Indeed, <math>f</math> is continuous because it is a composition of continuous functions. The restriction <math>\tan\colon (-\pi/2,\pi/2)\to\mathbb{R}</math> is bijective with continuous inverse <math>\arctan\colon \mathbb{R}\to(-\pi/2,\pi/2)</math>. Therefore <math>f</math> is bijective and its inverse<br />
<math display="block">f^{-1}(y)=\dfrac1\pi\arctan(y)+\dfrac12</math><br />
is continuous. Thus <math>f</math> is a homeomorphism.}}<br />
<br />
=== Stereographic projection ===<br />
The [[Euclidean plane]] <math>\mathbb{R}^2</math> is homeomorphic to the [[2-sphere]] minus one point, denoted <math>S^2 \setminus \{N\}</math> where <math>N=(0,0,1)</math> is the [[north pole]].<br />
<br />
{{Proof|proof=<br />
<div style="float:right; width:320px; margin:0 0 0.5em 1em;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=15pt]{standalone}<br />
\usepackage{tikz-3dplot}<br />
\usetikzlibrary{calc, arrows.meta}<br />
\begin{document}<br />
\def\viewTheta{70}<br />
\def\viewPhi{20}<br />
\tdplotsetmaincoords{\viewTheta}{\viewPhi}<br />
<br />
\begin{tikzpicture}[tdplot_main_coords, scale=2, line cap=round, line join=round]<br />
<br />
\def\R{1}<br />
\coordinate (O) at (0,0,0);<br />
\coordinate (N) at (0,0,\R);<br />
<br />
\def\thetaS{60}<br />
\def\phiS{30}<br />
\pgfmathsetmacro{\px}{\R * sin(\thetaS) * cos(\phiS)}<br />
\pgfmathsetmacro{\py}{\R * sin(\thetaS) * sin(\phiS)}<br />
\pgfmathsetmacro{\pz}{\R * cos(\thetaS)}<br />
\coordinate (P) at (\px, \py, \pz);<br />
<br />
\pgfmathsetmacro{\ux}{\px / (1 - \pz)}<br />
\pgfmathsetmacro{\uy}{\py / (1 - \pz)}<br />
\coordinate (Pprime) at (\ux, \uy, 0);<br />
<br />
\pgfmathsetmacro{\eqFrontStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\eqBackStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqBackEnd}{\viewPhi+180}<br />
<br />
\pgfmathsetmacro{\cotViewTheta}{cos(\viewTheta)/sin(\viewTheta)}<br />
\pgfmathsetmacro{\cotThetaS}{cos(\thetaS)/sin(\thetaS)}<br />
\pgfmathsetmacro{\cosAlpha}{max(min(-\cotThetaS * \cotViewTheta, 1), -1)}<br />
\pgfmathsetmacro{\alpha}{acos(\cosAlpha)}<br />
<br />
\pgfmathsetmacro{\latFrontStart}{\viewPhi-180}<br />
\pgfmathsetmacro{\latFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\latBackStart}{\viewPhi}<br />
\pgfmathsetmacro{\latBackEnd}{\viewPhi+180}<br />
<br />
\draw[thick, black] (-1.2,0,0) -- (0,0,0);<br />
\draw[thick, black] (0,-3,0) -- (0,0,0);<br />
\draw[thick, black] (0,0,-1.2) -- (0,0,0);<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (2.2,0,0) node[anchor=north east]{$x$};<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (0,3.0,0) node[anchor=north west]{$y$};<br />
\draw[thick, dashed] (0,0,0) -- (N);<br />
\draw[thick, ->, >=Stealth] (N) -- (0,0,1.8) node[anchor=south]{$z$};<br />
\begin{scope}[tdplot_screen_coords]<br />
\shade[ball color=cyan, opacity=0.15] (0,0) circle (\R);<br />
\draw[cyan!60!blue, thick] (0,0) circle (\R);<br />
\end{scope}<br />
<br />
\tdplotdrawarc[cyan!60!blue, thick]{(O)}{\R}{\eqFrontStart}{\eqFrontEnd}{}{}<br />
\tdplotdrawarc[cyan!60!blue, thin, dashed]{(O)}{\R}{\eqBackStart}{\eqBackEnd}{}{}<br />
<br />
\pgfmathsetmacro{\rLat}{\R * sin(\thetaS)}<br />
\coordinate (CenterLat) at (0,0,\pz);<br />
<br />
\tdplotsetrotatedcoords{0}{0}{0}<br />
\tdplotsetrotatedcoordsorigin{(CenterLat)}<br />
\tdplotdrawarc[cyan!70!black, thin, dashed, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latBackStart}{\latBackEnd}{}{}<br />
\tdplotdrawarc[cyan!70!black, thin, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latFrontStart}{\latFrontEnd}{}{}<br />
\tdplotsetrotatedcoordsorigin{(O)}<br />
<br />
\draw[red, thick, dashed] (N) -- (P);<br />
\draw[red, thick, ->, >=Stealth] (P) -- (Pprime);<br />
<br />
\fill[black] (N) circle (0.8pt) node[anchor=south east] {$N$};<br />
\fill[red] (P) circle (1pt) node[anchor=south west, text=black] {$(x,y,z)$};<br />
\fill[red] (Pprime) circle (1pt) node[anchor=north west, text=black] {$p(x,y,z) = (u,v)$};<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
Define the [[stereographic projection]] <math>p\colon S^2 \setminus \{N\} \to \mathbb{R}^2</math> by<br />
<math display="block">p(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right).</math><br />
This map is continuous because it is a rational function with denominator nonzero (since <math>z<1</math> on <math>S^2\setminus\{N\}</math>).<br />
<br />
The inverse map <math>p^{-1}\colon \mathbb{R}^2 \to S^2 \setminus \{N\}</math> is given by<br />
<math display="block">p^{-1}(u,v) = \left( \frac{2u}{u^2+v^2+1}, \frac{2v}{u^2+v^2+1}, \frac{u^2+v^2-1}{u^2+v^2+1} \right).</math><br />
This is also continuous as a composition of continuous functions. One verifies that <math>p \circ p^{-1} = \text{id}_{\mathbb{R}^2}</math> and <math>p^{-1} \circ p = \operatorname{id}_{S^2\setminus\{N\}}</math> by direct substitution. Hence <math>p</math> is a homeomorphism.<br />
}}<br />
<br />
=== Quotient space ===<br />
The unit interval <math>[0,1]</math> with the endpoints identified (the quotient space <math>[0,1]/\sim</math> where <math>0\sim 1</math>) is homeomorphic to the circle <math>S^1</math>.<br />
<br />
{{Proof|proof=Define the map <math>f\colon [0,1] \to S^1</math> by <math display="block">f(t)=(\cos(2\pi t), \sin(2\pi t)).</math> This map is continuous and [[Surjection|surjective]], and satisfies <math>f(0)=f(1)=(1,0)</math>.<br />
<br />
Consider the equivalence relation <math>\sim</math>, and let <math>q\colon [0,1]\to [0,1]/\sim</math> be the [[quotient map]]. By the [[universal property]] of the quotient map, there exists a unique continuous map <math>\tilde{f}\colon [0,1]/\sim \to S^1</math> such that <math>\tilde{f} \circ q = f</math>; that is, the following diagram commutes:<br />
<div style="text-align: center;"><br />
<br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
{[0,1]} && {S^1} \\<br />
& {[0,1]/{\sim}} \arrow["f", from=1-1, to=1-3]<br />
\arrow["q"', from=1-1, to=2-2]<br />
\arrow["{\exists! \tilde{f}}"', dashed, from=2-2, to=1-3]<br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
The map <math>\tilde{f}</math> is bijective because:<br />
* Surjectivity follows from surjectivity of <math>f</math>;<br />
* [[Injection|Injectivity]] holds because <math display="block">\tilde{f}([t])=\tilde{f}([s])\Rightarrow t=s \text{ or } \{t,s\}=\{0,1\},</math> but in the latter case <math>[t]=[s]</math> in the quotient. <br />
<br />
Hence <math>\tilde{f}</math> is a continuous bijection.<br />
<br />
The space <math>[0,1]/\sim</math> is compact as the quotient of a [[compact space]], and <math>S^1</math> is [[Hausdorff space|Hausdorff]]. By the [[Compact-to-Hausdorff theorem]], a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.<br />
<br />
Therefore <math>\tilde{f}</math> is a homeomorphism.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz">\documentclass[tikz,border=10pt]{standalone}<br />
\usepackage{amsmath}<br />
\usetikzlibrary{arrows.meta,calc}<br />
\begin{document}<br />
<br />
\begin{tikzpicture}[<br />
>={Stealth[scale=1.1]},<br />
dot/.style={circle,fill=black,inner sep=1.6pt},<br />
label text/.style={font=\Large,align=center}<br />
]<br />
<br />
\def\r{1.4}<br />
\def\gap{50}<br />
<br />
\coordinate (C1) at (0,0);<br />
\coordinate (C2) at (5.5,0);<br />
\coordinate (C3) at (11,0);<br />
<br />
\begin{scope}[shift={(C1)}]<br />
\draw[thick] (-\r,0) coordinate (A) -- (\r,0) coordinate (B);<br />
\node[dot] at (A) {};<br />
\node[dot] at (B) {};<br />
\node[label text] at (0,-2.6) {$[0,1]$};<br />
\coordinate (R1) at (\r,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C1)+(2,0)$) -- ($(C2)+(-2,0)$)<br />
node[midway,above] {$q$};<br />
<br />
\begin{scope}[shift={(C2)}]<br />
\draw[thick]<br />
(180-\gap:\r)<br />
arc[start angle=180-\gap,end angle=360+\gap,radius=\r];<br />
<br />
\node[dot] (L) at (180-\gap:\r) {};<br />
\node[dot] (R) at (\gap:\r) {};<br />
<br />
\draw[dashed,->,thick]<br />
(L) .. controls +(0,0) and +(-0.8,-0.1) .. (90:\r);<br />
\draw[dashed,->,thick]<br />
(R) .. controls +(0,0) and +(0.8,-0.1) .. (90:\r);<br />
<br />
\node[label text] at (0,-2.6)<br />
{$[0,1]/\sim$ \\[-0.4ex]\normalsize $(0\sim1)$};<br />
<br />
\coordinate (R2) at (2,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C2)+(2,0)$) -- ($(C3)+(-2,0)$)<br />
node[midway,above] {$\overset{\tilde{f}}{\cong}$};<br />
<br />
\begin{scope}[shift={(C3)}]<br />
\draw[thick] (0,0) circle (\r);<br />
\node[dot] at (90:\r) {};<br />
\node[label text] at (0,-2.6) {$S^{1}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}</kroki><br />
</div><br />
<br />
}}<br />
<br />
== Counterexamples ==<br />
<math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>.<br />
<br />
{{Proof|proof=For contradiction, suppose that there exists a homeomorphism <math>f\colon \mathbb{R}\to\mathbb{R}^2</math>.<br />
<br />
Consider the subspace <math>\mathbb{R}\setminus\{0\}</math> of <math>\mathbb{R}</math>. The [[restriction]] on it, <math>\left.f\right|_{\mathbb{R}\setminus\{0\}}\colon \mathbb{R}\setminus\{0\}\to \mathbb{R}^2\setminus\{f(0)\}</math> is also a homeomorphism.<br />
<br />
However, <math>\mathbb{R}\setminus\{0\}</math> has two connected components, <math>(-\infty,0)</math> and <math>(0,\infty)</math>, while <math>\mathbb{R}^2\setminus\{f(0)\}</math> is connected, which contradicts the assumption that the two spaces are homeomorphic.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz, border=10pt]{standalone}<br />
\usepackage{amsmath, amssymb}<br />
<br />
\begin{document}<br />
\begin{tikzpicture}<br />
\begin{scope}[xshift=-5cm]<br />
\draw[thick] (-3, 0) -- (3, 0);<br />
<br />
\filldraw[fill=white, draw=black, thick] (0, 0) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2.5, 0) -- (-1, 0);<br />
\fill[cyan!60!blue] (-2.5, 0) circle (2pt);<br />
\fill[cyan!60!blue] (-1, 0) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (1.3, 0) -- (2.2, 0);<br />
\fill[cyan!60!blue] (1.3, 0) circle (2pt);<br />
\fill[cyan!60!blue] (2.2, 0) circle (2pt);<br />
<br />
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<br />
Hence, no such homeomorphism exists; therefore <math>\mathbb{R}</math> is not homeomorphic to <math>\mathbb{R}^2</math>}}<br />
<br />
The map from the interval <math>[0,1)</math> to the 1-sphere <math>S^1</math>,<br />
<math display="block">\phi\colon [0,1)\to S^1,\quad x\mapsto e^{2\pi ix}</math><br />
is continuous and bijective, but not a homeomorphism. <br />
{{Proof|proof=<br />
<div style="text-align: center;"><br />
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The map <math>\phi</math> is:<br />
* Continuous, as it is the composition of continuous maps <math>x\mapsto 2\pi x</math> and <math>t\mapsto e^{it}</math>.<br />
* Injective, because if <math>e^{2\pi i x_1}=e^{2\pi i x_2}</math>, then <math>x_1-x_2\in \mathbb{Z}</math>. Since <math>x_1,x_2\in [0,1)</math>, it follows that <math>x_1=x_2</math>.<br />
* Surjective, since every point of <math>S^1</math> can be written as <math>e^{2\pi i x}</math> for some <math>x\in [0,1)</math>.<br />
Hence <math>\phi</math> is a continuous bijection.<br />
<br />
However, <math>\phi</math> is not a homeomorphism.<br />
<br />
Consider the sequence<math display="block">z_n = e^{2\pi i (1-\tfrac{1}{n})} \in S^1.</math><br />
Then <math>z_n \to 1 = e^{2\pi i \cdot 0}</math> in <math>S^1</math>. But<br />
<math display="block">\phi^{-1}(z_n) = 1-\frac{1}{n} \to 1,</math><br />
which does not converge to <math>\phi^{-1}(1)=0</math> in <math>[0,1)</math>. Thus <math>\phi^{-1}</math> is not continuous.<br />
}}<br />
<br />
== Topological invariants ==<br />
A [[topological invariant]] is a property of a topological space that is preserved under homeomorphisms. In other words, if two spaces are homeomorphic, they either both possess the property or both do not. Invariants are the important tools to classify topological spaces. If two spaces differ in any topological invariant, they cannot be homeomorphic. Conversely, showing that two spaces share many invariants is often the first step on proving they are homeomorpic, though it is never sufficient by itself.<br />
<br />
=== Common topological invariants ===<br />
<br />
* [[Connectedness]]<br />
* [[Compactness]]<br />
* [[Hausdorff space|Hausdorff property]]<br />
* [[Cardinality]] of the space<br />
<br />
=== Algebraic invariants ===<br />
More powerful invariants come from [[algebraic topology]], which assigns algebraic objects to topological spaces.<br />
<br />
* [[Fundamental group]]<br />
* [[Homology group]]<br />
* [[Higher homotopy group]]<br />
<br />
==Homeomorphism group==<br />
The collection of all [[autohomeomorphisms]] of a topological space <math>X</math> forms a [[group]] under composition operation, known as the homeomorphism group of <math>X</math>, denoted <math>\operatorname{Homeo}(X)</math>. The homeomorphism group captures the symmetry in topology. It describes the ways in which a topological space can be continuously transformed onto itself.<br />
<br />
The homeomorphism group <math>\operatorname{Homeo}(X)</math> is a faithful [[group action]] on its underlying set <math>X</math>. It moves points in <math>X</math> continuously onto <math>X</math> itself, and the topological structure of <math>X</math> is also reflected in the algebraic invariants such as the [[Orbit|orbits]] and [[Stabilizer|stabilizers]] of the action.<br />
<br />
For example, consider the 2-sphere <math>S^2</math> as a thin rubber membrane tightly wraped around a ball. Each autohomeomorphism of <math>S^2</math>, which is an element in <math>\operatorname{Homeo}(S^2)</math>, corresponds to a continuous deformation of this membrane. This operation can be stretching, bending, twisting, or any composition of these operations, so the rubber always remains attached to the ball.<br />
<div style="text-align: center;"><br />
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Under the natural action of <math>\operatorname{Homeo}(S^2)</math>, every point on the sphere can be moved continuously to any other point. This example shows how the homeomorphism group captures the symmetry of a topological space in the perspective of continuity.<br />
<br />
==See also==<br />
* [[Homotopy]]<br />
* [[Topology]]<br />
* [[Homeomorphism group]]<br />
<br />
==Terminology==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homeomorphism | homéomorphisme | Homeomorphismus | 同胚 | 同胚 | 同相写像 }}<br />
{{Terminology_table/row | homeomorphic | homéomorphe | homeomorph | 同胚的 | 同胚的 | 同相 }}<br />
{{Terminology_table/row | topological invariant | invariant topologique | topologische Invariante | 拓扑不变量 | 拓撲不變量 | 位相不変量 }}<br />
{{Terminology_table/row | autohomeomorphism | autohoméomorphisme | Selbsthomöomorphismus | 自同胚 | 自同胚 | 自己同相写像 }}<br />
{{Terminology_table/row | homeomorphism group | groupe des homéomorphismes | Homöomorphismengruppe | 同胚群 | 同胚群 | 同相群 }}<br />
}}<br />
<br />
[[Category:Topology]]</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Topological_space&diff=108Topological space2026-04-12T06:26:23Z<p>InfernalAtom683: </p>
<hr />
<div>A '''topological space''' is a fundamental mathematical structure that generalizes the concept of geometrical spaces and [[continuity]]. A topological space is equipped with a collection of [[open sets]], capturing the intuitive idea of "nearness" without necessarily defining a [[metric]]. Topological spaces are the objects of study in [[general topology]].<br />
<br />
== Definition ==<br />
An [[ordered pair]] <math>(X, \tau)</math> is a topological space on set <math>X</math>, if <math>\tau\subseteq \mathcal{P}(X)</math> satisfies the following properties:<br />
<br />
* <math>X, \varnothing \in \tau</math>,<br />
* if <math>\mathcal{U}\subseteq \tau</math>, then <math>\bigcup_{U\in \mathcal{U}}U \in \tau</math>,<br />
* if <math>U_1, U_2, \cdots, U_n\in \tau</math>, then <math>\bigcap_{i=1}^n U_i \in \tau</math>.<br />
<br />
Elements of <math>\tau</math> are called [[Open set|open sets]].<br />
<br />
== Examples ==<br />
<br />
=== Standard topology ===<br />
The real line <math>\mathbb{R}</math> equipped with the standard topology <math>\tau_{\mathbb{R}}</math> is a topological space. <br />
<br />
The standard topology on <math>\mathbb{R}</math> is defined by taking all open intervals as a [[basis]]. A set <math>U\subseteq \mathbb{R}</math> is open, if for all point <math>x\in U</math>, there exists an open interval <math>(a,b)</math> such that <math>x\in (a,b)\subseteq U</math>.<br />
{{Proof|proof=<br />
* <math>\varnothing\in\tau_{\mathbb{R}}</math> vacuously; every point <math>x\in\mathbb{R}</math> belongs to some open interval, like <math>(x-1,x+1)</math>, which is open in <math>\mathbb{R}</math>. Therefore by the definition, <math>\mathbb{R}\in\tau_{\mathbb{R}}</math>.<br />
* Let <math>\{U_i\}_{i\in I}</math> be open sets and <math>U=\bigcup_{i\in I}U_i</math>. Take <math>x\in U</math>, then <math>x\in U_i</math> for some <math>i\in I</math>. Because <math>U_i</math> is open, there exists <math>(a,b)</math> such that <math>x\in (a,b)\subset U_i\subset U</math>. Therefore <math>U</math> is open.<br />
* The finite intersection property can be proved by induction. Let <math>U, V</math> be open and <math>x\in U\cap V</math>. By definition of openness, there exists <math>(a_1,b_1)\subset U</math> and <math>(a_2,b_2)\subset V</math> such that <math>(a_1,b_1)\ni x \in (a_2,b_2)</math>. Set <math>a=\max\{a_1,a_2\}</math> and <math>b=\min\{b_1,b_2\}</math>. Then <math>x\in(a,b)\subset (a_1,b_1)\cap(a_2,b_2)\subset U\cap V</math>. Thus <math>U\cap V</math> is open. By induction, <math></math><br />
}}<br />
<br />
<br />
<br />
==Basis==<br />
* for every <math>x \in X</math>, there exists <math>B \in \mathcal{B}</math> with <math>x \in B</math>,<br />
* if <math>x \in B_1 \cap B_2</math> with <math>B_1, B_2 \in \mathcal{B}</math>, then there exists <math>B_3 \in \mathcal{B}</math> such that <math>x \in B_3 \subseteq B_1 \cap B_2</math>.<br />
<br />
The topology generated by <math>\mathcal{B}</math> consists of all unions of elements of <math>\mathcal{B}</math>.<br />
<br />
== Topological properties ==<br />
Some key properties of topological spaces include:<br />
* [[Compact space|Compactness]]<br />
* [[Connected space|Connectedness]]<br />
* [[Hausdorff space|Hausdorff property]]<br />
* [[Second-countable space|Second countability]]</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Topological_space&diff=107Topological space2026-04-12T06:26:19Z<p>InfernalAtom683: </p>
<hr />
<div>A '''topological space''' is a fundamental mathematical structure that generalizes the concept of geometrical spaces and [[continuity]]. A topological space is equipped with a collection of [[open sets]], capturing the intuitive idea of "nearness" without necessarily defining a [[metric]]. Topological spaces are the objects of study in [[general topology]].<br />
<br />
== Definition ==<br />
An [[ordered pair]] <math>(X, \tau)</math> is a topological space on set <math>X</math>, if <math>\tau\subseteq \mathcal{P}(X)</math> is a '''topology''', satisfying the following properties:<br />
<br />
* <math>X, \varnothing \in \tau</math>,<br />
* if <math>\mathcal{U}\subseteq \tau</math>, then <math>\bigcup_{U\in \mathcal{U}}U \in \tau</math>,<br />
* if <math>U_1, U_2, \cdots, U_n\in \tau</math>, then <math>\bigcap_{i=1}^n U_i \in \tau</math>.<br />
<br />
Elements of <math>\tau</math> are called [[Open set|open sets]].<br />
<br />
== Examples ==<br />
<br />
=== Standard topology ===<br />
The real line <math>\mathbb{R}</math> equipped with the '''standard topology''' <math>\tau_{\mathbb{R}}</math> is a topological space. <br />
<br />
The standard topology on <math>\mathbb{R}</math> is defined by taking all open intervals as a [[basis]]. A set <math>U\subseteq \mathbb{R}</math> is open, if for all point <math>x\in U</math>, there exists an open interval <math>(a,b)</math> such that <math>x\in (a,b)\subseteq U</math>.<br />
{{Proof|proof=<br />
* <math>\varnothing\in\tau_{\mathbb{R}}</math> vacuously; every point <math>x\in\mathbb{R}</math> belongs to some open interval, like <math>(x-1,x+1)</math>, which is open in <math>\mathbb{R}</math>. Therefore by the definition, <math>\mathbb{R}\in\tau_{\mathbb{R}}</math>.<br />
* Let <math>\{U_i\}_{i\in I}</math> be open sets and <math>U=\bigcup_{i\in I}U_i</math>. Take <math>x\in U</math>, then <math>x\in U_i</math> for some <math>i\in I</math>. Because <math>U_i</math> is open, there exists <math>(a,b)</math> such that <math>x\in (a,b)\subset U_i\subset U</math>. Therefore <math>U</math> is open.<br />
* Let <math>U, V</math> be open and <math>x\in U\cap V</math>. By definition of openness, there exists <math>(a_1,b_1)\subset U</math> and <math>(a_2,b_2)\subset V</math> such that <math>(a_1,b_1)\ni x \in (a_2,b_2)</math>. Set <math>a=\max\{a_1,a_2\}</math> and <math>b=\min\{b_1,b_2\}</math>. Then <math>x\in(a,b)\subset (a_1,b_1)\cap(a_2,b_2)\subset U\cap V</math>. Thus <math>U\cap V</math> is open. By induction, the finite intersection property holds.<br />
<br />
Therefore, <math>\tau_\mathbb{R}</math> is indeed a topology on <math>\mathbb{R}</math>.<br />
}}<br />
<br />
=== Discrete topology ===<br />
Let <math>X</math> be an arbitary set and define the '''discrete topology''' of <math>X</math> by <math>\tau=\mathcal{P}(X)</math>. Every subset of <math>X</math> is open in <math>\tau</math>.<br />
{{Proof|proof=* <math>\emptyset, X\subseteq X</math>, thus <math>\emptyset, X\in \tau</math>.<br />
* Let <math>X\supseteq A,B\in \tau</math>, since <math>A</math> and <math>B</math> are subsets of <math>X</math>, their union <math>A\cup B</math> and intersection <math>A\cap B</math> are also a subsets of <math>X</math>, which are in the topology <math>\tau</math>.<br />
<br />
Therefore the discrete topology of <math>X</math> is a topology.}}<br />
<br />
=== Indiscrete topology ===<br />
Let <math>X</math> be an arbitary set, the '''indiscrete topology''' of <math>X</math> is defined by <math>\tau=\{\emptyset, X\}</math>.<br />
{{Proof|proof=* By definition, <math>\emptyset, X\in \tau</math>.<br />
* <math>\emptyset \cup X=X\in\tau</math>.<br />
* <math>\emptyset \cap X=\emptyset\in\tau</math>.<br />
<br />
Therefore the indiscrete topology of <math>X</math> is a topology.}}<br />
<br />
==Basis==<br />
* for every <math>x \in X</math>, there exists <math>B \in \mathcal{B}</math> with <math>x \in B</math>,<br />
* if <math>x \in B_1 \cap B_2</math> with <math>B_1, B_2 \in \mathcal{B}</math>, then there exists <math>B_3 \in \mathcal{B}</math> such that <math>x \in B_3 \subseteq B_1 \cap B_2</math>.<br />
<br />
The topology generated by <math>\mathcal{B}</math> consists of all unions of elements of <math>\mathcal{B}</math>.<br />
<br />
== Topological properties ==<br />
Some key properties of topological spaces include:<br />
* [[Compact space|Compactness]]<br />
* [[Connected space|Connectedness]]<br />
* [[Hausdorff space|Hausdorff property]]<br />
* [[Second-countable space|Second countability]]</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Compact_space&diff=106Compact space2026-04-11T22:56:00Z<p>InfernalAtom683: </p>
<hr />
<div>A '''compact''' [[topological space]] is one that behaves, in many respects, like a finite space, even if it is infinite. Specifically, a compact space is a topological space whose every open cover admits a finite subcover. Compactness is one of the most fundamental [[Topological property|topological properties]] in [[analysis]] and [[topology]].<br />
<br />
Intuitively, compactness can be understood as a generalization of being "[[Closed set|closed]] and [[Boundedness|bounded]]". In [[Euclidean space|Euclidean spaces]] <math>\mathbb{R}^n</math>, by the [[Heine–Borel theorem]], a set is compact if and only if it is closed and bounded.<br />
<br />
However, in a general topological space, a [[metric]] is typically not available, thus "boundedness" cannot be defined in a meaningful way. Therefore, an adopted definition is the one using open cover. In <math>\mathbb{R}^n</math>, this condition is equivalent to being closed and bounded, while still making sense in arbitrary topological spaces and preserving the essential properties of compact sets.<br />
<br />
== Definition ==<br />
A topological space <math>X</math> is compact if for every collection <math>\{U_i\}_{i\in I}</math> of opensets in <math>X</math> such that<br />
<math display="block">\bigcup_{i\in I}U_i,</math><br />
there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,U_{i_n}\}</math> such that<br />
<math display="block">\bigcup_{k=1}^n U_{i_k}.</math></div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Compact_space&diff=105Compact space2026-04-11T22:55:26Z<p>InfernalAtom683: </p>
<hr />
<div>A '''compact''' [[topological space]] is one that behaves, in many respects, like a finite space, even if it is infinite. Specifically, a compact space is a topological space whose every open cover admits a finite subcover. Compactness is one of the most fundamental [[Topological property|topological properties]] in [[analysis]] and [[topology]].<br />
<br />
Intuitively, compactness can be understood as a generalization of being "[[Closed set|closed]] and [[Boundedness|bounded]]". In [[Euclidean space|Euclidean spaces]] <math>\mathbb{R}^n</math>, by the [[Heine–Borel theorem]], a set is compact if and only if it is closed and bounded.<br />
<br />
However, in a general topological space, a [[metric]] is typically not available, thus "boundedness" cannot be defined in a meaningful way. Therefore, an adopted definition is the one using open cover. In <math>\mathbb{R}^n</math>, this condition is equivalent to being closed and bounded, while still making sense in arbitrary topological spaces and preserving the essential properties of compact sets.<br />
<br />
== Definition ==<br />
A topological space <math>X</math> is compact if for every collection <math>\{U_i\}_{i\in I}</math> of opensets in <math>X</math> such that<br />
<math display="block">\bigcup_{i\in I}U_i,</math><br />
there exists a finite subcollection <math>\{U_{i_1},U_{i_2},\dots,\U_{i_n}\}</math> such that<br />
<math display="block">\bigcup_{k=1}^n U_{i_k}.</math></div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homeomorphism&diff=104Homeomorphism2026-04-11T07:49:01Z<p>InfernalAtom683: </p>
<hr />
<div>[[File:Topology joke.jpg|thumb|250x250px|A homeomorphism that turns a coffee mug into a donut continuously.]]<br />
A '''homeomorphism''' is a special type of [[function]] between two [[Topological space|topological spaces]], that establishes that the two spaces are fundamentally the same from a topological perspective. Specifically, it is a [[Continuous function|continuous]] [[bijective]] function whose [[inverse function]] is also continuous. Homeomorphisms are the [[Isomorphism|isomorphisms]] in the [[category of topological spaces]] <math>\mathsf{Top}</math>, which preserves all [[topological properties]] of a topological space. If such a function exists between two spaces, they are said to be '''homeomorphic'''. <br />
<br />
Intuitively, two spaces are homeomorphic if one can be continuously deformed into the other by stretching, bending, and twisting, without cutting, tearing, or gluing. A typical intuitive example is that a mug with a handle is homeomorphic to a donut. This concept is distinct from [[Homotopy#Homotopy equivalence|homotopy equivalence]], which allows deformations that involve collapsing. For instance, a solid ball can be continuously shrunk to a point by a homotopy, but such a deformation is not a homeomorphism because it is not bijective and the inverse would not be continuous.<br />
<br />
== Definitions ==<br />
A function <math>f</math> between topological spaces <math>X</math> and <math>Y</math> is called a '''homeomorphism''', if:<br />
<br />
* <math>f</math> is continuous,<br />
* <math>f</math> is bijective,<br />
* <math>f^{-1}</math> is continuous.<br />
<br />
Two topological spaces <math>X</math> and <math>Y</math> are called '''homeomorphic''' if there exists a homeomorphism between them, denoted <math>X\cong Y</math>.<br />
<br />
=== Equivalent Definitions ===<br />
A homeomorphism is a bijection that is continuous and [[Open function|open]], or continuous and [[Closed function|closed]].<br />
<br />
== Properties ==<br />
{{Property|property=The composition of two homeomorphisms is again a homeomorphism.}}{{Proof|proof=Let <math>f\colon X \to Y</math> and <math>g\colon Y \to Z</math> be homeomorphisms. Then:<br />
<br />
* <math>g \circ f\colon X \to Z</math> is bijective, since the composition of two bijections is a bijection.<br />
<br />
* <math>g \circ f</math> is continuous, as the composition of two continuous functions.<br />
<br />
* The inverse is <math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}</math>, which is continuous because it is the composition of the continuous functions <math>g^{-1}</math> and <math>f^{-1}</math>.<br />
<br />
Thus <math>g \circ f</math> satisfies all requirements of a homeomorphism.}}<br />
<br />
<br />
{{Property|property=The inverse of a homeomorphism is again a homeomorphism.}}<br />
{{Proof|proof=Let <math>f\colon X \to Y</math> be a homeomorphism. Then:<br />
* <math>f^{-1}</math> is continuous by definition,<br />
* <math>f^{-1}</math> is bijective, since the inverse of a bijection is again a bijection,<br />
* <math>\left(f^{-1}\right)^{-1}=f</math> is continuous by definition.}}<br />
<br />
<br />
{{Property|property=Homeomorphism is an [[equivalence relation]].}}<br />
<br />
{{Proof|proof=<br />
* '''Reflexivity''': The identity map <math>\operatorname{id}_X\colon X\to X</math> is a continuous bijection on any topological space <math>X</math>, whose inverse is itself. Thus <math>\operatorname{id}_X</math> is a homeomorphism.<br />
* '''Symmetry''': If <math>f\colon X\to Y</math> is a homeomorphism, then its inverse <math>f^{-1}\colon Y\to X</math> is again a homeomorphism.<br />
* '''Transitivity''': If <math>f\colon X\to Y</math> and <math>g\colon Y\to Z</math> are homeomorphisms, then <math>g\circ f: X\to Z</math> is again a homeomorphism.<br />
}}<br />
<br />
== Examples ==<br />
<br />
=== Open interval ===<br />
The [[open interval]] <math>(0,1)</math> is homeomorphic to <math>\mathbb{R}</math>.<br />
{{Proof|proof=The map <math>f\colon (0,1)\to \mathbb{R}</math> defined by<br />
<math display="block">f(x)=\tan\left(\pi\left(x-\dfrac12\right)\right)</math><br />
is a homeomorphism. Indeed, <math>f</math> is continuous because it is a composition of continuous functions. The restriction <math>\tan\colon (-\pi/2,\pi/2)\to\mathbb{R}</math> is bijective with continuous inverse <math>\arctan\colon \mathbb{R}\to(-\pi/2,\pi/2)</math>. Therefore <math>f</math> is bijective and its inverse<br />
<math display="block">f^{-1}(y)=\dfrac1\pi\arctan(y)+\dfrac12</math><br />
is continuous. Thus <math>f</math> is a homeomorphism.}}<br />
<br />
=== Stereographic projection ===<br />
The [[Euclidean plane]] <math>\mathbb{R}^2</math> is homeomorphic to the [[2-sphere]] minus one point, denoted <math>S^2 \setminus \{N\}</math> where <math>N=(0,0,1)</math> is the [[north pole]].<br />
<br />
{{Proof|proof=<br />
<div style="float:right; width:320px; margin:0 0 0.5em 1em;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=15pt]{standalone}<br />
\usepackage{tikz-3dplot}<br />
\usetikzlibrary{calc, arrows.meta}<br />
\begin{document}<br />
\def\viewTheta{70}<br />
\def\viewPhi{20}<br />
\tdplotsetmaincoords{\viewTheta}{\viewPhi}<br />
<br />
\begin{tikzpicture}[tdplot_main_coords, scale=2, line cap=round, line join=round]<br />
<br />
\def\R{1}<br />
\coordinate (O) at (0,0,0);<br />
\coordinate (N) at (0,0,\R);<br />
<br />
\def\thetaS{60}<br />
\def\phiS{30}<br />
\pgfmathsetmacro{\px}{\R * sin(\thetaS) * cos(\phiS)}<br />
\pgfmathsetmacro{\py}{\R * sin(\thetaS) * sin(\phiS)}<br />
\pgfmathsetmacro{\pz}{\R * cos(\thetaS)}<br />
\coordinate (P) at (\px, \py, \pz);<br />
<br />
\pgfmathsetmacro{\ux}{\px / (1 - \pz)}<br />
\pgfmathsetmacro{\uy}{\py / (1 - \pz)}<br />
\coordinate (Pprime) at (\ux, \uy, 0);<br />
<br />
\pgfmathsetmacro{\eqFrontStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\eqBackStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqBackEnd}{\viewPhi+180}<br />
<br />
\pgfmathsetmacro{\cotViewTheta}{cos(\viewTheta)/sin(\viewTheta)}<br />
\pgfmathsetmacro{\cotThetaS}{cos(\thetaS)/sin(\thetaS)}<br />
\pgfmathsetmacro{\cosAlpha}{max(min(-\cotThetaS * \cotViewTheta, 1), -1)}<br />
\pgfmathsetmacro{\alpha}{acos(\cosAlpha)}<br />
<br />
\pgfmathsetmacro{\latFrontStart}{\viewPhi-180}<br />
\pgfmathsetmacro{\latFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\latBackStart}{\viewPhi}<br />
\pgfmathsetmacro{\latBackEnd}{\viewPhi+180}<br />
<br />
\draw[thick, black] (-1.2,0,0) -- (0,0,0);<br />
\draw[thick, black] (0,-3,0) -- (0,0,0);<br />
\draw[thick, black] (0,0,-1.2) -- (0,0,0);<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (2.2,0,0) node[anchor=north east]{$x$};<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (0,3.0,0) node[anchor=north west]{$y$};<br />
\draw[thick, dashed] (0,0,0) -- (N);<br />
\draw[thick, ->, >=Stealth] (N) -- (0,0,1.8) node[anchor=south]{$z$};<br />
\begin{scope}[tdplot_screen_coords]<br />
\shade[ball color=cyan, opacity=0.15] (0,0) circle (\R);<br />
\draw[cyan!60!blue, thick] (0,0) circle (\R);<br />
\end{scope}<br />
<br />
\tdplotdrawarc[cyan!60!blue, thick]{(O)}{\R}{\eqFrontStart}{\eqFrontEnd}{}{}<br />
\tdplotdrawarc[cyan!60!blue, thin, dashed]{(O)}{\R}{\eqBackStart}{\eqBackEnd}{}{}<br />
<br />
\pgfmathsetmacro{\rLat}{\R * sin(\thetaS)}<br />
\coordinate (CenterLat) at (0,0,\pz);<br />
<br />
\tdplotsetrotatedcoords{0}{0}{0}<br />
\tdplotsetrotatedcoordsorigin{(CenterLat)}<br />
\tdplotdrawarc[cyan!70!black, thin, dashed, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latBackStart}{\latBackEnd}{}{}<br />
\tdplotdrawarc[cyan!70!black, thin, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latFrontStart}{\latFrontEnd}{}{}<br />
\tdplotsetrotatedcoordsorigin{(O)}<br />
<br />
\draw[red, thick, dashed] (N) -- (P);<br />
\draw[red, thick, ->, >=Stealth] (P) -- (Pprime);<br />
<br />
\fill[black] (N) circle (0.8pt) node[anchor=south east] {$N$};<br />
\fill[red] (P) circle (1pt) node[anchor=south west, text=black] {$(x,y,z)$};<br />
\fill[red] (Pprime) circle (1pt) node[anchor=north west, text=black] {$p(x,y,z) = (u,v)$};<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
Define the [[stereographic projection]] <math>p\colon S^2 \setminus \{N\} \to \mathbb{R}^2</math> by<br />
<math display="block">p(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right).</math><br />
This map is continuous because it is a rational function with denominator nonzero (since <math>z<1</math> on <math>S^2\setminus\{N\}</math>).<br />
<br />
The inverse map <math>p^{-1}\colon \mathbb{R}^2 \to S^2 \setminus \{N\}</math> is given by<br />
<math display="block">p^{-1}(u,v) = \left( \frac{2u}{u^2+v^2+1}, \frac{2v}{u^2+v^2+1}, \frac{u^2+v^2-1}{u^2+v^2+1} \right).</math><br />
This is also continuous as a composition of continuous functions. One verifies that <math>p \circ p^{-1} = \text{id}_{\mathbb{R}^2}</math> and <math>p^{-1} \circ p = \operatorname{id}_{S^2\setminus\{N\}}</math> by direct substitution. Hence <math>p</math> is a homeomorphism.<br />
}}<br />
<br />
=== Quotient space ===<br />
The unit interval <math>[0,1]</math> with the endpoints identified (the quotient space <math>[0,1]/\sim</math> where <math>0\sim 1</math>) is homeomorphic to the circle <math>S^1</math>.<br />
<br />
{{Proof|proof=Define the map <math>f\colon [0,1] \to S^1</math> by <math display="block">f(t)=(\cos(2\pi t), \sin(2\pi t)).</math> This map is continuous and [[Surjection|surjective]], and satisfies <math>f(0)=f(1)=(1,0)</math>.<br />
<br />
Consider the equivalence relation <math>\sim</math>, and let <math>q\colon [0,1]\to [0,1]/\sim</math> be the [[quotient map]]. By the [[universal property]] of the quotient map, there exists a unique continuous map <math>\tilde{f}\colon [0,1]/\sim \to S^1</math> such that <math>\tilde{f} \circ q = f</math>; that is, the following diagram commutes.<br />
<div style="text-align: center;"><br />
<br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
{[0,1]} && {S^1} \\<br />
& {[0,1]/{\sim}} \arrow["f", from=1-1, to=1-3]<br />
\arrow["q"', from=1-1, to=2-2]<br />
\arrow["{\exists! \tilde{f}}"', dashed, from=2-2, to=1-3]<br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
The map <math>\tilde{f}</math> is bijective because:<br />
* Surjectivity follows from surjectivity of <math>f</math>;<br />
* [[Injection|Injectivity]] holds because <math display="block">\tilde{f}([t])=\tilde{f}([s])\Rightarrow t=s \text{ or } \{t,s\}=\{0,1\},</math> but in the latter case <math>[t]=[s]</math> in the quotient. <br />
<br />
Hence <math>\tilde{f}</math> is a continuous bijection.<br />
<br />
The space <math>[0,1]/\sim</math> is compact as the quotient of a [[compact space]], and <math>S^1</math> is [[Hausdorff space|Hausdorff]]. By the [[Compact-to-Hausdorff theorem]], a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.<br />
<br />
Therefore <math>\tilde{f}</math> is a homeomorphism.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz">\documentclass[tikz,border=10pt]{standalone}<br />
\usepackage{amsmath}<br />
\usetikzlibrary{arrows.meta,calc}<br />
\begin{document}<br />
<br />
\begin{tikzpicture}[<br />
>={Stealth[scale=1.1]},<br />
dot/.style={circle,fill=black,inner sep=1.6pt},<br />
label text/.style={font=\Large,align=center}<br />
]<br />
<br />
\def\r{1.4}<br />
\def\gap{50}<br />
<br />
\coordinate (C1) at (0,0);<br />
\coordinate (C2) at (5.5,0);<br />
\coordinate (C3) at (11,0);<br />
<br />
\begin{scope}[shift={(C1)}]<br />
\draw[thick] (-\r,0) coordinate (A) -- (\r,0) coordinate (B);<br />
\node[dot] at (A) {};<br />
\node[dot] at (B) {};<br />
\node[label text] at (0,-2.6) {$[0,1]$};<br />
\coordinate (R1) at (\r,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C1)+(2,0)$) -- ($(C2)+(-2,0)$)<br />
node[midway,above] {$q$};<br />
<br />
\begin{scope}[shift={(C2)}]<br />
\draw[thick]<br />
(180-\gap:\r)<br />
arc[start angle=180-\gap,end angle=360+\gap,radius=\r];<br />
<br />
\node[dot] (L) at (180-\gap:\r) {};<br />
\node[dot] (R) at (\gap:\r) {};<br />
<br />
\draw[dashed,->,thick]<br />
(L) .. controls +(0,0) and +(-0.8,-0.1) .. (90:\r);<br />
\draw[dashed,->,thick]<br />
(R) .. controls +(0,0) and +(0.8,-0.1) .. (90:\r);<br />
<br />
\node[label text] at (0,-2.6)<br />
{$[0,1]/\sim$ \\[-0.4ex]\normalsize $(0\sim1)$};<br />
<br />
\coordinate (R2) at (2,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C2)+(2,0)$) -- ($(C3)+(-2,0)$)<br />
node[midway,above] {$\overset{\tilde{f}}{\cong}$};<br />
<br />
\begin{scope}[shift={(C3)}]<br />
\draw[thick] (0,0) circle (\r);<br />
\node[dot] at (90:\r) {};<br />
\node[label text] at (0,-2.6) {$S^{1}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}</kroki><br />
</div><br />
<br />
}}<br />
<br />
== Counterexamples ==<br />
<math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>.<br />
<br />
{{Proof|proof=For contradiction, suppose that there exists a homeomorphism <math>f\colon \mathbb{R}\to\mathbb{R}^2</math>.<br />
<br />
Consider the subspace <math>\mathbb{R}\setminus\{0\}</math> of <math>\mathbb{R}</math>. The [[restriction]] on it, <math>\left.f\right|_{\mathbb{R}\setminus\{0\}}\colon \mathbb{R}\setminus\{0\}\to \mathbb{R}^2\setminus\{f(0)\}</math> is also a homeomorphism.<br />
<br />
However, <math>\mathbb{R}\setminus\{0\}</math> has two connected components, <math>(-\infty,0)</math> and <math>(0,\infty)</math>, while <math>\mathbb{R}^2\setminus\{f(0)\}</math> is connected, which contradicts the assumption that the two spaces are homeomorphic.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz, border=10pt]{standalone}<br />
\usepackage{amsmath, amssymb}<br />
<br />
\begin{document}<br />
\begin{tikzpicture}<br />
\begin{scope}[xshift=-5cm]<br />
\draw[thick] (-3, 0) -- (3, 0);<br />
<br />
\filldraw[fill=white, draw=black, thick] (0, 0) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2.5, 0) -- (-1, 0);<br />
\fill[cyan!60!blue] (-2.5, 0) circle (2pt);<br />
\fill[cyan!60!blue] (-1, 0) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (1.3, 0) -- (2.2, 0);<br />
\fill[cyan!60!blue] (1.3, 0) circle (2pt);<br />
\fill[cyan!60!blue] (2.2, 0) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -2) {$\mathbb{R} \setminus \{0\}$};<br />
\end{scope}<br />
<br />
\begin{scope}[xshift=4cm]<br />
\draw[thick] (-3.5, -2.5) -- (3.5, -2.5) -- (3.5, 2.5) -- (-3.5, 2.5) -- cycle;<br />
<br />
\filldraw[fill=white, draw=black, thick] (-0.3, -0.2) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2, 1.5) .. controls (-0.5, 1) and (-0.8, -1) .. (-1.2, -1.8);<br />
\fill[cyan!60!blue] (-2, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (-1.2, -1.8) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 1.5) .. controls (1.5, 1.8) and (2.5, 1) .. (2, 0.5);<br />
\fill[cyan!60!blue] (-0.3, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (2, 0.5) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 0.5) .. controls (1, 0) and (1, -1) .. (2.5, -2);<br />
\fill[cyan!60!blue] (-0.3, 0.5) circle (2pt);<br />
\fill[cyan!60!blue] (2.5, -2) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.5, -0.8) .. controls (0, -1.8) and (1, -1.5) .. (0.8, -1);<br />
\fill[cyan!60!blue] (-0.5, -0.8) circle (2pt);<br />
\fill[cyan!60!blue] (0.8, -1) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -3.5) {$\mathbb{R}^2 \setminus \{f(0)\}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
Hence, no such homeomorphism exists; therefore <math>\mathbb{R}</math> is not homeomorphic to <math>\mathbb{R}^2</math>}}<br />
<br />
The map from the interval <math>[0,1)</math> to the 1-sphere <math>S^1</math>,<br />
<math display="block">\phi\colon [0,1)\to S^1,\quad x\mapsto e^{2\pi ix}</math><br />
is continuous and bijective, but not a homeomorphism. <br />
{{Proof|proof=<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass{article}<br />
\usepackage{tikz}<br />
\usetikzlibrary{arrows.meta}<br />
<br />
\begin{document}<br />
<br />
\begin{tikzpicture}<br />
\draw (0,0) -- (4,0);<br />
<br />
\draw (0.15, 0.25) -- (0, 0.25) -- (0, -0.25) -- (0.15, -0.25);<br />
<br />
\draw (3.9, 0.25) to[bend left=45] (3.9, -0.25);<br />
<br />
\draw[-{Stealth[length=3mm, width=2mm]}] (4.5, 1.2) to[bend left=30] node[above=2pt] {$\phi$} (7.0, 1.2);<br />
<br />
\draw (9.5, 0) circle (2);<br />
<br />
\begin{scope}[rotate around={90:(11.5,0)}]<br />
\draw (11.39, 0.25) to[bend left=45] (11.39, -0.25);<br />
\draw (11.5, -0.25) -- (11.5, 0.25);<br />
\draw (11.5, 0.25) -- (11.65, 0.25);<br />
\draw (11.5, -0.25) -- (11.65, -0.25);<br />
<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
The map <math>\phi</math> is:<br />
* Continuous, as it is the composition of continuous maps <math>x\mapsto 2\pi x</math> and <math>t\mapsto e^{it}</math>.<br />
* Injective, because if <math>e^{2\pi i x_1}=e^{2\pi i x_2}</math>, then <math>x_1-x_2\in \mathbb{Z}</math>. Since <math>x_1,x_2\in [0,1)</math>, it follows that <math>x_1=x_2</math>.<br />
* Surjective, since every point of <math>S^1</math> can be written as <math>e^{2\pi i x}</math> for some <math>x\in [0,1)</math>.<br />
Hence <math>\phi</math> is a continuous bijection.<br />
<br />
However, <math>\phi</math> is not a homeomorphism.<br />
<br />
Consider the sequence<math display="block">z_n = e^{2\pi i (1-\tfrac{1}{n})} \in S^1.</math><br />
Then <math>z_n \to 1 = e^{2\pi i \cdot 0}</math> in <math>S^1</math>. But<br />
<math display="block">\phi^{-1}(z_n) = 1-\frac{1}{n} \to 1,</math><br />
which does not converge to <math>\phi^{-1}(1)=0</math> in <math>[0,1)</math>. Thus <math>\phi^{-1}</math> is not continuous.<br />
}}<br />
<br />
== Topological invariants ==<br />
A [[topological invariant]] is a property of a topological space that is preserved under homeomorphisms. In other words, if two spaces are homeomorphic, they either both possess the property or both do not. Invariants are the important tools to classify topological spaces. If two spaces differ in any topological invariant, they cannot be homeomorphic. Conversely, showing that two spaces share many invariants is often the first step on proving they are homeomorpic, though it is never sufficient by itself.<br />
<br />
=== Common topological invariants ===<br />
<br />
* [[Connectedness]]<br />
* [[Compactness]]<br />
* [[Hausdorff space|Hausdorff property]]<br />
* [[Cardinality]] of the space<br />
<br />
=== Algebraic invariants ===<br />
More powerful invariants come from [[algebraic topology]], which assigns algebraic objects to topological spaces.<br />
<br />
* [[Fundamental group]]<br />
* [[Homology group]]<br />
* [[Higher homotopy group]]<br />
<br />
==Homeomorphism group==<br />
The collection of all [[autohomeomorphisms]] of a topological space <math>X</math> forms a [[group]] under composition operation, known as the homeomorphism group of <math>X</math>, denoted <math>\operatorname{Homeo}(X)</math>. The homeomorphism group captures the symmetry in topology. It describes the ways in which a topological space can be continuously transformed onto itself.<br />
<br />
The homeomorphism group <math>\operatorname{Homeo}(X)</math> is a faithful [[group action]] on its underlying set <math>X</math>. It moves points in <math>X</math> continuously onto <math>X</math> itself, and the topological structure of <math>X</math> is also reflected in the algebraic invariants such as the [[Orbit|orbits]] and [[Stabilizer|stabilizers]] of the action.<br />
<br />
For example, consider the 2-sphere <math>S^2</math> as a thin rubber membrane tightly wraped around a ball. Each autohomeomorphism of <math>S^2</math>, which is an element in <math>\operatorname{Homeo}(S^2)</math>, corresponds to a continuous deformation of this membrane. This operation can be stretching, bending, twisting, or any composition of these operations, so the rubber always remains attached to the ball.<br />
<div style="text-align: center;"><br />
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\draw[cyan!70!black, thin] (0,2) -- (0,-2);<br />
<br />
\filldraw[fill=yellow!60, draw=yellow!60!black, thick, opacity=0.7]<br />
(0.7, 1.1) .. controls (1.0, 1.3) and (1.3, 1.1) .. (1.4, 0.5) <br />
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<br />
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(0.05, 1.3) <br />
.. controls (0.4, 1.4) and (0.9, 1.5) .. (1.2, 0.7) <br />
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</kroki><br />
</div><br />
Under the natural action of <math>\operatorname{Homeo}(S^2)</math>, every point on the sphere can be moved continuously to any other point. This example shows how the homeomorphism group captures the symmetry of a topological space in the perspective of continuity.<br />
<br />
==See also==<br />
* [[Homotopy]]<br />
* [[Topology]]<br />
* [[Homeomorphism group]]<br />
<br />
==Terminology==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homeomorphism | homéomorphisme | Homeomorphismus | 同胚 | 同胚 | 同相写像 }}<br />
{{Terminology_table/row | homeomorphic | homéomorphe | homeomorph | 同胚的 | 同胚的 | 同相 }}<br />
{{Terminology_table/row | topological invariant | invariant topologique | topologische Invariante | 拓扑不变量 | 拓撲不變量 | 位相不変量 }}<br />
{{Terminology_table/row | autohomeomorphism | autohoméomorphisme | Selbsthomöomorphismus | 自同胚 | 自同胚 | 自己同相写像 }}<br />
{{Terminology_table/row | homeomorphism group | groupe des homéomorphismes | Homöomorphismengruppe | 同胚群 | 同胚群 | 同相群 }}<br />
}}<br />
<br />
[[Category:Topology]]</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homeomorphism&diff=103Homeomorphism2026-04-11T07:47:15Z<p>InfernalAtom683: /* Counterexamples */</p>
<hr />
<div>[[File:Topology joke.jpg|thumb|250x250px|A homeomorphism that turns a coffee mug into a donut continuously.]]<br />
A '''homeomorphism''' is a special type of [[function]] between two [[Topological space|topological spaces]], that establishes that the two spaces are fundamentally the same from a topological perspective. Specifically, it is a [[Continuous function|continuous]] [[bijective]] function whose [[inverse function]] is also continuous. Homeomorphisms are the [[Isomorphism|isomorphisms]] in the [[category of topological spaces]] <math>\mathsf{Top}</math>, which preserves all [[topological properties]] of a topological space. If such a function exists between two spaces, they are said to be '''homeomorphic'''. <br />
<br />
Intuitively, two spaces are homeomorphic if one can be continuously deformed into the other by stretching, bending, and twisting, without cutting, tearing, or gluing. A typical intuitive example is that a mug with a handle is homeomorphic to a donut. This concept is distinct from [[Homotopy#Homotopy equivalence|homotopy equivalence]], which allows deformations that involve collapsing. For instance, a solid ball can be continuously shrunk to a point by a homotopy, but such a deformation is not a homeomorphism because it is not bijective and the inverse would not be continuous.<br />
<br />
== Definitions ==<br />
A function <math>f</math> between topological spaces <math>X</math> and <math>Y</math> is called a '''homeomorphism''', if:<br />
<br />
* <math>f</math> is continuous,<br />
* <math>f</math> is bijective,<br />
* <math>f^{-1}</math> is continuous.<br />
<br />
Two topological spaces <math>X</math> and <math>Y</math> are called '''homeomorphic''' if there exists a homeomorphism between them, denoted <math>X\cong Y</math>.<br />
<br />
=== Equivalent Definitions ===<br />
A homeomorphism is a bijection that is continuous and [[Open function|open]], or continuous and [[Closed function|closed]].<br />
<br />
== Properties ==<br />
{{Property|property=The composition of two homeomorphisms is again a homeomorphism.}}{{Proof|proof=Let <math>f\colon X \to Y</math> and <math>g\colon Y \to Z</math> be homeomorphisms. Then:<br />
<br />
* <math>g \circ f\colon X \to Z</math> is bijective, since the composition of two bijections is a bijection.<br />
<br />
* <math>g \circ f</math> is continuous, as the composition of two continuous functions.<br />
<br />
* The inverse is <math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}</math>, which is continuous because it is the composition of the continuous functions <math>g^{-1}</math> and <math>f^{-1}</math>.<br />
<br />
Thus <math>g \circ f</math> satisfies all requirements of a homeomorphism.}}<br />
<br />
<br />
{{Property|property=The inverse of a homeomorphism is again a homeomorphism.}}<br />
{{Proof|proof=Let <math>f\colon X \to Y</math> be a homeomorphism. Then:<br />
* <math>f^{-1}</math> is continuous by definition,<br />
* <math>f^{-1}</math> is bijective, since the inverse of a bijection is again a bijection,<br />
* <math>\left(f^{-1}\right)^{-1}=f</math> is continuous by definition.}}<br />
<br />
<br />
{{Property|property=Homeomorphism is an [[equivalence relation]].}}<br />
<br />
{{Proof|proof=<br />
* '''Reflexivity''': The identity map <math>\operatorname{id}_X\colon X\to X</math> is a continuous bijection on any topological space <math>X</math>, whose inverse is itself. Thus <math>\operatorname{id}_X</math> is a homeomorphism.<br />
* '''Symmetry''': If <math>f\colon X\to Y</math> is a homeomorphism, then its inverse <math>f^{-1}\colon Y\to X</math> is again a homeomorphism.<br />
* '''Transitivity''': If <math>f\colon X\to Y</math> and <math>g\colon Y\to Z</math> are homeomorphisms, then <math>g\circ f: X\to Z</math> is again a homeomorphism.<br />
}}<br />
<br />
== Examples ==<br />
<br />
=== Open interval ===<br />
The [[open interval]] <math>(0,1)</math> is homeomorphic to <math>\mathbb{R}</math>.<br />
{{Proof|proof=The map <math>f\colon (0,1)\to \mathbb{R}</math> defined by<br />
<math display="block">f(x)=\tan\left(\pi\left(x-\dfrac12\right)\right)</math><br />
is a homeomorphism. Indeed, <math>f</math> is continuous because it is a composition of continuous functions. The restriction <math>\tan\colon (-\pi/2,\pi/2)\to\mathbb{R}</math> is bijective with continuous inverse <math>\arctan\colon \mathbb{R}\to(-\pi/2,\pi/2)</math>. Therefore <math>f</math> is bijective and its inverse<br />
<math display="block">f^{-1}(y)=\dfrac1\pi\arctan(y)+\dfrac12</math><br />
is continuous. Thus <math>f</math> is a homeomorphism.}}<br />
<br />
=== Stereographic projection ===<br />
The [[Euclidean plane]] <math>\mathbb{R}^2</math> is homeomorphic to the [[2-sphere]] minus one point, denoted <math>S^2 \setminus \{N\}</math> where <math>N=(0,0,1)</math> is the [[north pole]].<br />
<br />
{{Proof|proof=<br />
<div style="float:right; width:320px; margin:0 0 0.5em 1em;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=15pt]{standalone}<br />
\usepackage{tikz-3dplot}<br />
\usetikzlibrary{calc, arrows.meta}<br />
\begin{document}<br />
\def\viewTheta{70}<br />
\def\viewPhi{20}<br />
\tdplotsetmaincoords{\viewTheta}{\viewPhi}<br />
<br />
\begin{tikzpicture}[tdplot_main_coords, scale=2, line cap=round, line join=round]<br />
<br />
\def\R{1}<br />
\coordinate (O) at (0,0,0);<br />
\coordinate (N) at (0,0,\R);<br />
<br />
\def\thetaS{60}<br />
\def\phiS{30}<br />
\pgfmathsetmacro{\px}{\R * sin(\thetaS) * cos(\phiS)}<br />
\pgfmathsetmacro{\py}{\R * sin(\thetaS) * sin(\phiS)}<br />
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\coordinate (P) at (\px, \py, \pz);<br />
<br />
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\coordinate (Pprime) at (\ux, \uy, 0);<br />
<br />
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<br />
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<br />
\draw[red, thick, dashed] (N) -- (P);<br />
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<br />
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<br />
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<br />
\end{document}<br />
</kroki><br />
</div><br />
Define the [[stereographic projection]] <math>p\colon S^2 \setminus \{N\} \to \mathbb{R}^2</math> by<br />
<math display="block">p(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right).</math><br />
This map is continuous because it is a rational function with denominator nonzero (since <math>z<1</math> on <math>S^2\setminus\{N\}</math>).<br />
<br />
The inverse map <math>p^{-1}\colon \mathbb{R}^2 \to S^2 \setminus \{N\}</math> is given by<br />
<math display="block">p^{-1}(u,v) = \left( \frac{2u}{u^2+v^2+1}, \frac{2v}{u^2+v^2+1}, \frac{u^2+v^2-1}{u^2+v^2+1} \right).</math><br />
This is also continuous as a composition of continuous functions. One verifies that <math>p \circ p^{-1} = \text{id}_{\mathbb{R}^2}</math> and <math>p^{-1} \circ p = \operatorname{id}_{S^2\setminus\{N\}}</math> by direct substitution. Hence <math>p</math> is a homeomorphism.<br />
}}<br />
<br />
=== Quotient space ===<br />
The unit interval <math>[0,1]</math> with the endpoints identified (the quotient space <math>[0,1]/\sim</math> where <math>0\sim 1</math>) is homeomorphic to the circle <math>S^1</math>.<br />
<br />
{{Proof|proof=Define the map <math>f\colon [0,1] \to S^1</math> by <math display="block">f(t)=(\cos(2\pi t), \sin(2\pi t)).</math> This map is continuous and [[Surjection|surjective]], and satisfies <math>f(0)=f(1)=(1,0)</math>.<br />
<br />
Consider the equivalence relation <math>\sim</math>, and let <math>q\colon [0,1]\to [0,1]/\sim</math> be the [[quotient map]]. By the [[universal property]] of the quotient map, there exists a unique continuous map <math>\tilde{f}\colon [0,1]/\sim \to S^1</math> such that <math>\tilde{f} \circ q = f</math>; that is, the following diagram commutes.<br />
<div style="text-align: center;"><br />
<br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
{[0,1]} && {S^1} \\<br />
& {[0,1]/{\sim}} \arrow["f", from=1-1, to=1-3]<br />
\arrow["q"', from=1-1, to=2-2]<br />
\arrow["{\exists! \tilde{f}}"', dashed, from=2-2, to=1-3]<br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
The map <math>\tilde{f}</math> is bijective because:<br />
* Surjectivity follows from surjectivity of <math>f</math>;<br />
* [[Injection|Injectivity]] holds because <math display="block">\tilde{f}([t])=\tilde{f}([s])\Rightarrow t=s \text{ or } \{t,s\}=\{0,1\},</math> but in the latter case <math>[t]=[s]</math> in the quotient. <br />
<br />
Hence <math>\tilde{f}</math> is a continuous bijection.<br />
<br />
The space <math>[0,1]/\sim</math> is compact as the quotient of a [[compact space]], and <math>S^1</math> is [[Hausdorff space|Hausdorff]]. By the [[Compact-to-Hausdorff theorem]], a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.<br />
<br />
Therefore <math>\tilde{f}</math> is a homeomorphism.<br />
<br />
<div style="text-align: center;"><br />
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\usepackage{amsmath}<br />
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<br />
\def\r{1.4}<br />
\def\gap{50}<br />
<br />
\coordinate (C1) at (0,0);<br />
\coordinate (C2) at (5.5,0);<br />
\coordinate (C3) at (11,0);<br />
<br />
\begin{scope}[shift={(C1)}]<br />
\draw[thick] (-\r,0) coordinate (A) -- (\r,0) coordinate (B);<br />
\node[dot] at (A) {};<br />
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\node[label text] at (0,-2.6) {$[0,1]$};<br />
\coordinate (R1) at (\r,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C1)+(2,0)$) -- ($(C2)+(-2,0)$)<br />
node[midway,above] {$q$};<br />
<br />
\begin{scope}[shift={(C2)}]<br />
\draw[thick]<br />
(180-\gap:\r)<br />
arc[start angle=180-\gap,end angle=360+\gap,radius=\r];<br />
<br />
\node[dot] (L) at (180-\gap:\r) {};<br />
\node[dot] (R) at (\gap:\r) {};<br />
<br />
\draw[dashed,->,thick]<br />
(L) .. controls +(0,0) and +(-0.8,-0.1) .. (90:\r);<br />
\draw[dashed,->,thick]<br />
(R) .. controls +(0,0) and +(0.8,-0.1) .. (90:\r);<br />
<br />
\node[label text] at (0,-2.6)<br />
{$[0,1]/\sim$ \\[-0.4ex]\normalsize $(0\sim1)$};<br />
<br />
\coordinate (R2) at (2,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C2)+(2,0)$) -- ($(C3)+(-2,0)$)<br />
node[midway,above] {$\overset{\tilde{f}}{\cong}$};<br />
<br />
\begin{scope}[shift={(C3)}]<br />
\draw[thick] (0,0) circle (\r);<br />
\node[dot] at (90:\r) {};<br />
\node[label text] at (0,-2.6) {$S^{1}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}</kroki><br />
</div><br />
<br />
}}<br />
<br />
== Counterexamples ==<br />
<math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>.<br />
<br />
{{Proof|proof=For contradiction, suppose that there exists a homeomorphism <math>f\colon \mathbb{R}\to\mathbb{R}^2</math>.<br />
<br />
Consider the subspace <math>\mathbb{R}\setminus\{0\}</math> of <math>\mathbb{R}</math>. The [[restriction]] on it, <math>\left.f\right|_{\mathbb{R}\setminus\{0\}}\colon \mathbb{R}\setminus\{0\}\to \mathbb{R}^2\setminus\{f(0)\}</math> is also a homeomorphism.<br />
<br />
However, <math>\mathbb{R}\setminus\{0\}</math> has two connected components, <math>(-\infty,0)</math> and <math>(0,\infty)</math>, while <math>\mathbb{R}^2\setminus\{f(0)\}</math> is connected, which contradicts the assumption that the two spaces are homeomorphic.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
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\usepackage{amsmath, amssymb}<br />
<br />
\begin{document}<br />
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\begin{scope}[xshift=-5cm]<br />
\draw[thick] (-3, 0) -- (3, 0);<br />
<br />
\filldraw[fill=white, draw=black, thick] (0, 0) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2.5, 0) -- (-1, 0);<br />
\fill[cyan!60!blue] (-2.5, 0) circle (2pt);<br />
\fill[cyan!60!blue] (-1, 0) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (1.3, 0) -- (2.2, 0);<br />
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<br />
\draw[cyan!60!blue, very thick] (-0.5, -0.8) .. controls (0, -1.8) and (1, -1.5) .. (0.8, -1);<br />
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<br />
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\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
Hence, no such homeomorphism exists; therefore <math>\mathbb{R}</math> is not homeomorphic to <math>\mathbb{R}^2</math>}}<br />
<br />
The map from the interval <math>[0,1)</math> to the 1-sphere <math>S^1</math>,<br />
<math display="block">\phi\colon [0,1)\to S^1,\quad x\mapsto e^{2\pi ix}</math><br />
is continuous and bijective, but not a homeomorphism. <br />
{{Proof|proof=<br />
<div style="text-align: center;"><br />
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\draw (0,0) -- (4,0);<br />
<br />
\draw (0.15, 0.25) -- (0, 0.25) -- (0, -0.25) -- (0.15, -0.25);<br />
<br />
\draw (3.9, 0.25) to[bend left=45] (3.9, -0.25);<br />
<br />
\draw[-{Stealth[length=3mm, width=2mm]}] (4.5, 1.2) to[bend left=30] node[above=2pt] {$\phi$} (7.0, 1.2);<br />
<br />
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<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
The map <math>\phi</math> is:<br />
* Continuous, as it is the composition of continuous maps <math>x\mapsto 2\pi x</math> and <math>t\mapsto e^{it}</math>.<br />
* Injective, because if <math>e^{2\pi i x_1}=e^{2\pi i x_2}</math>, then <math>x_1-x_2\in \mathbb{Z}</math>. Since <math>x_1,x_2\in [0,1)</math>, it follows that <math>x_1=x_2</math>.<br />
* Surjective, since every point of <math>S^1</math> can be written as <math>e^{2\pi i x}</math> for some <math>x\in [0,1)</math>.<br />
Hence <math>\phi</math> is a continuous bijection.<br />
<br />
However, <math>\phi</math> is not a homeomorphism.<br />
<br />
Consider the sequence<math display="block">z_n = e^{2\pi i (1-\tfrac{1}{n})} \in S^1.</math><br />
Then <math>z_n \to 1 = e^{2\pi i \cdot 0}</math> in <math>S^1</math>. But<br />
<math display="block">\phi^{-1}(z_n) = 1-\frac{1}{n} \to 1,</math><br />
which does not converge to <math>\phi^{-1}(1)=0</math> in <math>[0,1)</math>. Thus <math>\phi^{-1}</math> is not continuous.<br />
}}<br />
<br />
== Topological invariants ==<br />
A [[topological invariant]] is a property of a topological space that is preserved under homeomorphisms. In other words, if two spaces are homeomorphic, they either both possess the property or both do not. Invariants are the important tools to classify topological spaces. If two spaces differ in any topological invariant, they cannot be homeomorphic. Conversely, showing that two spaces share many invariants is often the first step on proving they are homeomorpic, though it is never sufficient by itself.<br />
<br />
=== Common topological invariants ===<br />
<br />
* [[Connectedness]]<br />
* [[Compactness]]<br />
* [[Hausdorff space|Hausdorff property]]<br />
* [[Cardinality]] of the space<br />
<br />
=== Algebraic invariants ===<br />
More powerful invariants come from [[algebraic topology]], which assigns algebraic objects to topological spaces.<br />
<br />
* [[Fundamental group]]<br />
* [[Homology group]]<br />
* [[Higher homotopy group]]<br />
<br />
==Homeomorphism group==<br />
The collection of all [[autohomeomorphisms]] of a topological space <math>X</math> forms a [[group]] under composition operation, known as the homeomorphism group of <math>X</math>, denoted <math>\operatorname{Homeo}(X)</math>. The homeomorphism group captures the symmetry in topology. It describes the ways in which a topological space can be continuously transformed onto itself.<br />
<br />
The homeomorphism group <math>\operatorname{Homeo}(X)</math> is a faithful [[group action]] on its underlying set <math>X</math>. It moves points in <math>X</math> continuously onto <math>X</math> itself, and the topological structure of <math>X</math> is also reflected in the algebraic invariants such as the [[Orbit|orbits]] and [[Stabilizer|stabilizers]] of the action.<br />
<br />
For example, consider the 2-sphere <math>S^2</math> as a thin rubber membrane tightly wraped around a ball. Each autohomeomorphism of <math>S^2</math>, which is an element in <math>\operatorname{Homeo}(S^2)</math>, corresponds to a continuous deformation of this membrane. This operation can be stretching, bending, twisting, or any composition of these operations, so the rubber always remains attached to the ball.<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
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\usetikzlibrary{arrows.meta, positioning, calc}<br />
\usepackage{amssymb}<br />
\begin{document}<br />
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\begin{scope}[shift={(0,0)}]<br />
\shade[ball color=cyan, opacity=0.2] (0,0) circle (2);<br />
<br />
\draw[cyan!60!blue, thick] (-2,0) arc (180:360:2 and 0.25);<br />
\draw[cyan!60!blue, thin, dashed] (2,0) arc (0:180:2 and 0.25);<br />
<br />
\foreach \y in {-1.2, -0.6, 0.6, 1.2} {<br />
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\draw[cyan!70!black, thin, dashed] (\x, \y) arc (0:180:\x cm and 0.25cm);<br />
}<br />
<br />
\draw[cyan!70!black, thin] (0,2) arc (90:-90:0.6 and 2);<br />
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\draw[cyan!70!black, thin] (0,2) -- (0,-2);<br />
<br />
\filldraw[fill=yellow!60, draw=yellow!60!black, thick, opacity=0.7]<br />
(0.7, 1.1) .. controls (1.0, 1.3) and (1.3, 1.1) .. (1.4, 0.5) <br />
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<br />
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<br />
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<br />
\filldraw[fill=yellow!60, draw=yellow!60!black, thick, opacity=0.7]<br />
(0.05, 1.3) <br />
.. controls (0.4, 1.4) and (0.9, 1.5) .. (1.2, 0.7) <br />
.. controls (0.9, 0.3) and (0.4, 0.15) .. (0.05, 0.5) <br />
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<br />
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</div><br />
Under the natural action of <math>\operatorname{Homeo}(S^2)</math>, every point on the sphere can be moved continuously to any other point. This example shows how the homeomorphism group captures the symmetry of a topological space in the perspective of continuity.<br />
<br />
==See also==<br />
* [[Homotopy]]<br />
* [[Topology]]<br />
* [[Homeomorphism group]]<br />
<br />
[[Category:Topopogy]]<br />
<br />
==Terminology==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homeomorphism | homéomorphisme | Homeomorphismus | 同胚 | 同胚 | 同相写像 }}<br />
{{Terminology_table/row | homeomorphic | homéomorphe | homeomorph | 同胚的 | 同胚的 | 同相 }}<br />
{{Terminology_table/row | topological invariant | invariant topologique | topologische Invariante | 拓扑不变量 | 拓撲不變量 | 位相不変量 }}<br />
{{Terminology_table/row | autohomeomorphism | autohoméomorphisme | Selbsthomöomorphismus | 自同胚 | 自同胚 | 自己同相写像 }}<br />
{{Terminology_table/row | homeomorphism group | groupe des homéomorphismes | Homöomorphismengruppe | 同胚群 | 同胚群 | 同相群 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homeomorphism&diff=102Homeomorphism2026-04-11T07:22:30Z<p>InfernalAtom683: /* Counterexamples */</p>
<hr />
<div>[[File:Topology joke.jpg|thumb|250x250px|A homeomorphism that turns a coffee mug into a donut continuously.]]<br />
A '''homeomorphism''' is a special type of [[function]] between two [[Topological space|topological spaces]], that establishes that the two spaces are fundamentally the same from a topological perspective. Specifically, it is a [[Continuous function|continuous]] [[bijective]] function whose [[inverse function]] is also continuous. Homeomorphisms are the [[Isomorphism|isomorphisms]] in the [[category of topological spaces]] <math>\mathsf{Top}</math>, which preserves all [[topological properties]] of a topological space. If such a function exists between two spaces, they are said to be '''homeomorphic'''. <br />
<br />
Intuitively, two spaces are homeomorphic if one can be continuously deformed into the other by stretching, bending, and twisting, without cutting, tearing, or gluing. A typical intuitive example is that a mug with a handle is homeomorphic to a donut. This concept is distinct from [[Homotopy#Homotopy equivalence|homotopy equivalence]], which allows deformations that involve collapsing. For instance, a solid ball can be continuously shrunk to a point by a homotopy, but such a deformation is not a homeomorphism because it is not bijective and the inverse would not be continuous.<br />
<br />
== Definitions ==<br />
A function <math>f</math> between topological spaces <math>X</math> and <math>Y</math> is called a '''homeomorphism''', if:<br />
<br />
* <math>f</math> is continuous,<br />
* <math>f</math> is bijective,<br />
* <math>f^{-1}</math> is continuous.<br />
<br />
Two topological spaces <math>X</math> and <math>Y</math> are called '''homeomorphic''' if there exists a homeomorphism between them, denoted <math>X\cong Y</math>.<br />
<br />
=== Equivalent Definitions ===<br />
A homeomorphism is a bijection that is continuous and [[Open function|open]], or continuous and [[Closed function|closed]].<br />
<br />
== Properties ==<br />
{{Property|property=The composition of two homeomorphisms is again a homeomorphism.}}{{Proof|proof=Let <math>f\colon X \to Y</math> and <math>g\colon Y \to Z</math> be homeomorphisms. Then:<br />
<br />
* <math>g \circ f\colon X \to Z</math> is bijective, since the composition of two bijections is a bijection.<br />
<br />
* <math>g \circ f</math> is continuous, as the composition of two continuous functions.<br />
<br />
* The inverse is <math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}</math>, which is continuous because it is the composition of the continuous functions <math>g^{-1}</math> and <math>f^{-1}</math>.<br />
<br />
Thus <math>g \circ f</math> satisfies all requirements of a homeomorphism.}}<br />
<br />
<br />
{{Property|property=The inverse of a homeomorphism is again a homeomorphism.}}<br />
{{Proof|proof=Let <math>f\colon X \to Y</math> be a homeomorphism. Then:<br />
* <math>f^{-1}</math> is continuous by definition,<br />
* <math>f^{-1}</math> is bijective, since the inverse of a bijection is again a bijection,<br />
* <math>\left(f^{-1}\right)^{-1}=f</math> is continuous by definition.}}<br />
<br />
<br />
{{Property|property=Homeomorphism is an [[equivalence relation]].}}<br />
<br />
{{Proof|proof=<br />
* '''Reflexivity''': The identity map <math>\operatorname{id}_X\colon X\to X</math> is a continuous bijection on any topological space <math>X</math>, whose inverse is itself. Thus <math>\operatorname{id}_X</math> is a homeomorphism.<br />
* '''Symmetry''': If <math>f\colon X\to Y</math> is a homeomorphism, then its inverse <math>f^{-1}\colon Y\to X</math> is again a homeomorphism.<br />
* '''Transitivity''': If <math>f\colon X\to Y</math> and <math>g\colon Y\to Z</math> are homeomorphisms, then <math>g\circ f: X\to Z</math> is again a homeomorphism.<br />
}}<br />
<br />
== Examples ==<br />
<br />
=== Open interval ===<br />
The [[open interval]] <math>(0,1)</math> is homeomorphic to <math>\mathbb{R}</math>.<br />
{{Proof|proof=The map <math>f\colon (0,1)\to \mathbb{R}</math> defined by<br />
<math display="block">f(x)=\tan\left(\pi\left(x-\dfrac12\right)\right)</math><br />
is a homeomorphism. Indeed, <math>f</math> is continuous because it is a composition of continuous functions. The restriction <math>\tan\colon (-\pi/2,\pi/2)\to\mathbb{R}</math> is bijective with continuous inverse <math>\arctan\colon \mathbb{R}\to(-\pi/2,\pi/2)</math>. Therefore <math>f</math> is bijective and its inverse<br />
<math display="block">f^{-1}(y)=\dfrac1\pi\arctan(y)+\dfrac12</math><br />
is continuous. Thus <math>f</math> is a homeomorphism.}}<br />
<br />
=== Stereographic projection ===<br />
The [[Euclidean plane]] <math>\mathbb{R}^2</math> is homeomorphic to the [[2-sphere]] minus one point, denoted <math>S^2 \setminus \{N\}</math> where <math>N=(0,0,1)</math> is the [[north pole]].<br />
<br />
{{Proof|proof=<br />
<div style="float:right; width:320px; margin:0 0 0.5em 1em;"><br />
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\def\viewTheta{70}<br />
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\tdplotsetmaincoords{\viewTheta}{\viewPhi}<br />
<br />
\begin{tikzpicture}[tdplot_main_coords, scale=2, line cap=round, line join=round]<br />
<br />
\def\R{1}<br />
\coordinate (O) at (0,0,0);<br />
\coordinate (N) at (0,0,\R);<br />
<br />
\def\thetaS{60}<br />
\def\phiS{30}<br />
\pgfmathsetmacro{\px}{\R * sin(\thetaS) * cos(\phiS)}<br />
\pgfmathsetmacro{\py}{\R * sin(\thetaS) * sin(\phiS)}<br />
\pgfmathsetmacro{\pz}{\R * cos(\thetaS)}<br />
\coordinate (P) at (\px, \py, \pz);<br />
<br />
\pgfmathsetmacro{\ux}{\px / (1 - \pz)}<br />
\pgfmathsetmacro{\uy}{\py / (1 - \pz)}<br />
\coordinate (Pprime) at (\ux, \uy, 0);<br />
<br />
\pgfmathsetmacro{\eqFrontStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\eqBackStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqBackEnd}{\viewPhi+180}<br />
<br />
\pgfmathsetmacro{\cotViewTheta}{cos(\viewTheta)/sin(\viewTheta)}<br />
\pgfmathsetmacro{\cotThetaS}{cos(\thetaS)/sin(\thetaS)}<br />
\pgfmathsetmacro{\cosAlpha}{max(min(-\cotThetaS * \cotViewTheta, 1), -1)}<br />
\pgfmathsetmacro{\alpha}{acos(\cosAlpha)}<br />
<br />
\pgfmathsetmacro{\latFrontStart}{\viewPhi-180}<br />
\pgfmathsetmacro{\latFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\latBackStart}{\viewPhi}<br />
\pgfmathsetmacro{\latBackEnd}{\viewPhi+180}<br />
<br />
\draw[thick, black] (-1.2,0,0) -- (0,0,0);<br />
\draw[thick, black] (0,-3,0) -- (0,0,0);<br />
\draw[thick, black] (0,0,-1.2) -- (0,0,0);<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (2.2,0,0) node[anchor=north east]{$x$};<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (0,3.0,0) node[anchor=north west]{$y$};<br />
\draw[thick, dashed] (0,0,0) -- (N);<br />
\draw[thick, ->, >=Stealth] (N) -- (0,0,1.8) node[anchor=south]{$z$};<br />
\begin{scope}[tdplot_screen_coords]<br />
\shade[ball color=cyan, opacity=0.15] (0,0) circle (\R);<br />
\draw[cyan!60!blue, thick] (0,0) circle (\R);<br />
\end{scope}<br />
<br />
\tdplotdrawarc[cyan!60!blue, thick]{(O)}{\R}{\eqFrontStart}{\eqFrontEnd}{}{}<br />
\tdplotdrawarc[cyan!60!blue, thin, dashed]{(O)}{\R}{\eqBackStart}{\eqBackEnd}{}{}<br />
<br />
\pgfmathsetmacro{\rLat}{\R * sin(\thetaS)}<br />
\coordinate (CenterLat) at (0,0,\pz);<br />
<br />
\tdplotsetrotatedcoords{0}{0}{0}<br />
\tdplotsetrotatedcoordsorigin{(CenterLat)}<br />
\tdplotdrawarc[cyan!70!black, thin, dashed, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latBackStart}{\latBackEnd}{}{}<br />
\tdplotdrawarc[cyan!70!black, thin, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latFrontStart}{\latFrontEnd}{}{}<br />
\tdplotsetrotatedcoordsorigin{(O)}<br />
<br />
\draw[red, thick, dashed] (N) -- (P);<br />
\draw[red, thick, ->, >=Stealth] (P) -- (Pprime);<br />
<br />
\fill[black] (N) circle (0.8pt) node[anchor=south east] {$N$};<br />
\fill[red] (P) circle (1pt) node[anchor=south west, text=black] {$(x,y,z)$};<br />
\fill[red] (Pprime) circle (1pt) node[anchor=north west, text=black] {$p(x,y,z) = (u,v)$};<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
Define the [[stereographic projection]] <math>p\colon S^2 \setminus \{N\} \to \mathbb{R}^2</math> by<br />
<math display="block">p(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right).</math><br />
This map is continuous because it is a rational function with denominator nonzero (since <math>z<1</math> on <math>S^2\setminus\{N\}</math>).<br />
<br />
The inverse map <math>p^{-1}\colon \mathbb{R}^2 \to S^2 \setminus \{N\}</math> is given by<br />
<math display="block">p^{-1}(u,v) = \left( \frac{2u}{u^2+v^2+1}, \frac{2v}{u^2+v^2+1}, \frac{u^2+v^2-1}{u^2+v^2+1} \right).</math><br />
This is also continuous as a composition of continuous functions. One verifies that <math>p \circ p^{-1} = \text{id}_{\mathbb{R}^2}</math> and <math>p^{-1} \circ p = \operatorname{id}_{S^2\setminus\{N\}}</math> by direct substitution. Hence <math>p</math> is a homeomorphism.<br />
}}<br />
<br />
=== Quotient space ===<br />
The unit interval <math>[0,1]</math> with the endpoints identified (the quotient space <math>[0,1]/\sim</math> where <math>0\sim 1</math>) is homeomorphic to the circle <math>S^1</math>.<br />
<br />
{{Proof|proof=Define the map <math>f\colon [0,1] \to S^1</math> by <math display="block">f(t)=(\cos(2\pi t), \sin(2\pi t)).</math> This map is continuous and [[Surjection|surjective]], and satisfies <math>f(0)=f(1)=(1,0)</math>.<br />
<br />
Consider the equivalence relation <math>\sim</math>, and let <math>q\colon [0,1]\to [0,1]/\sim</math> be the [[quotient map]]. By the [[universal property]] of the quotient map, there exists a unique continuous map <math>\tilde{f}\colon [0,1]/\sim \to S^1</math> such that <math>\tilde{f} \circ q = f</math>; that is, the following diagram commutes.<br />
<div style="text-align: center;"><br />
<br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
{[0,1]} && {S^1} \\<br />
& {[0,1]/{\sim}} \arrow["f", from=1-1, to=1-3]<br />
\arrow["q"', from=1-1, to=2-2]<br />
\arrow["{\exists! \tilde{f}}"', dashed, from=2-2, to=1-3]<br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
The map <math>\tilde{f}</math> is bijective because:<br />
* Surjectivity follows from surjectivity of <math>f</math>;<br />
* [[Injection|Injectivity]] holds because <math display="block">\tilde{f}([t])=\tilde{f}([s])\Rightarrow t=s \text{ or } \{t,s\}=\{0,1\},</math> but in the latter case <math>[t]=[s]</math> in the quotient. <br />
<br />
Hence <math>\tilde{f}</math> is a continuous bijection.<br />
<br />
The space <math>[0,1]/\sim</math> is compact as the quotient of a [[compact space]], and <math>S^1</math> is [[Hausdorff space|Hausdorff]]. By the [[Compact-to-Hausdorff theorem]], a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.<br />
<br />
Therefore <math>\tilde{f}</math> is a homeomorphism.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz">\documentclass[tikz,border=10pt]{standalone}<br />
\usepackage{amsmath}<br />
\usetikzlibrary{arrows.meta,calc}<br />
\begin{document}<br />
<br />
\begin{tikzpicture}[<br />
>={Stealth[scale=1.1]},<br />
dot/.style={circle,fill=black,inner sep=1.6pt},<br />
label text/.style={font=\Large,align=center}<br />
]<br />
<br />
\def\r{1.4}<br />
\def\gap{50}<br />
<br />
\coordinate (C1) at (0,0);<br />
\coordinate (C2) at (5.5,0);<br />
\coordinate (C3) at (11,0);<br />
<br />
\begin{scope}[shift={(C1)}]<br />
\draw[thick] (-\r,0) coordinate (A) -- (\r,0) coordinate (B);<br />
\node[dot] at (A) {};<br />
\node[dot] at (B) {};<br />
\node[label text] at (0,-2.6) {$[0,1]$};<br />
\coordinate (R1) at (\r,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C1)+(2,0)$) -- ($(C2)+(-2,0)$)<br />
node[midway,above] {$q$};<br />
<br />
\begin{scope}[shift={(C2)}]<br />
\draw[thick]<br />
(180-\gap:\r)<br />
arc[start angle=180-\gap,end angle=360+\gap,radius=\r];<br />
<br />
\node[dot] (L) at (180-\gap:\r) {};<br />
\node[dot] (R) at (\gap:\r) {};<br />
<br />
\draw[dashed,->,thick]<br />
(L) .. controls +(0,0) and +(-0.8,-0.1) .. (90:\r);<br />
\draw[dashed,->,thick]<br />
(R) .. controls +(0,0) and +(0.8,-0.1) .. (90:\r);<br />
<br />
\node[label text] at (0,-2.6)<br />
{$[0,1]/\sim$ \\[-0.4ex]\normalsize $(0\sim1)$};<br />
<br />
\coordinate (R2) at (2,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C2)+(2,0)$) -- ($(C3)+(-2,0)$)<br />
node[midway,above] {$\overset{\tilde{f}}{\cong}$};<br />
<br />
\begin{scope}[shift={(C3)}]<br />
\draw[thick] (0,0) circle (\r);<br />
\node[dot] at (90:\r) {};<br />
\node[label text] at (0,-2.6) {$S^{1}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}</kroki><br />
</div><br />
<br />
}}<br />
<br />
== Counterexamples ==<br />
<math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>.<br />
<br />
{{Proof|proof=For contradiction, suppose that there exists a homeomorphism <math>f\colon \mathbb{R}\to\mathbb{R}^2</math>.<br />
<br />
Consider the subspace <math>\mathbb{R}\setminus\{0\}</math> of <math>\mathbb{R}</math>. The [[restriction]] on it, <math>\left.f\right|_{\mathbb{R}\setminus\{0\}}\colon \mathbb{R}\setminus\{0\}\to \mathbb{R}^2\setminus\{f(0)\}</math> is also a homeomorphism.<br />
<br />
However, <math>\mathbb{R}\setminus\{0\}</math> has two connected components, <math>(-\infty,0)</math> and <math>(0,\infty)</math>, while <math>\mathbb{R}^2\setminus\{f(0)\}</math> is connected, which contradicts the assumption that the two spaces are homeomorphic.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz, border=10pt]{standalone}<br />
\usepackage{amsmath, amssymb}<br />
<br />
\begin{document}<br />
\begin{tikzpicture}<br />
\begin{scope}[xshift=-5cm]<br />
\draw[thick] (-3, 0) -- (3, 0);<br />
<br />
\filldraw[fill=white, draw=black, thick] (0, 0) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2.5, 0) -- (-1, 0);<br />
\fill[cyan!60!blue] (-2.5, 0) circle (2pt);<br />
\fill[cyan!60!blue] (-1, 0) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (1.3, 0) -- (2.2, 0);<br />
\fill[cyan!60!blue] (1.3, 0) circle (2pt);<br />
\fill[cyan!60!blue] (2.2, 0) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -2) {$\mathbb{R} \setminus \{0\}$};<br />
\end{scope}<br />
<br />
\begin{scope}[xshift=4cm]<br />
\draw[thick] (-3.5, -2.5) -- (3.5, -2.5) -- (3.5, 2.5) -- (-3.5, 2.5) -- cycle;<br />
<br />
\filldraw[fill=white, draw=black, thick] (-0.3, -0.2) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2, 1.5) .. controls (-0.5, 1) and (-0.8, -1) .. (-1.2, -1.8);<br />
\fill[cyan!60!blue] (-2, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (-1.2, -1.8) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 1.5) .. controls (1.5, 1.8) and (2.5, 1) .. (2, 0.5);<br />
\fill[cyan!60!blue] (-0.3, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (2, 0.5) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 0.5) .. controls (1, 0) and (1, -1) .. (2.5, -2);<br />
\fill[cyan!60!blue] (-0.3, 0.5) circle (2pt);<br />
\fill[cyan!60!blue] (2.5, -2) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.5, -0.8) .. controls (0, -1.8) and (1, -1.5) .. (0.8, -1);<br />
\fill[cyan!60!blue] (-0.5, -0.8) circle (2pt);<br />
\fill[cyan!60!blue] (0.8, -1) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -3.5) {$\mathbb{R}^2 \setminus \{f(0)\}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
Hence, no such homeomorphism exists; therefore <math>\mathbb{R}</math> is not homeomorphic to <math>\mathbb{R}^2</math>}}<br />
<br />
The map from the interval <math>[0,1)</math> to the 1-sphere <math>S^1</math>,<br />
<math display="block">\phi\colon [0,1)\to S^1,\quad x\mapsto e^{2\pi ix}</math><br />
is continuous and bijective, but not a homeomorphism. <br />
{{Proof|proof=<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass{article}<br />
\usepackage{tikz}<br />
\usetikzlibrary{arrows.meta}<br />
<br />
\begin{document}<br />
<br />
\begin{tikzpicture}<br />
\draw (0,0) -- (4,0);<br />
<br />
\draw (0.15, 0.25) -- (0, 0.25) -- (0, -0.25) -- (0.15, -0.25);<br />
<br />
\draw (3.9, 0.25) to[bend left=45] (3.9, -0.25);<br />
<br />
\draw[-{Stealth[length=3mm, width=2mm]}] (4.5, 1.2) to[bend left=30] node[above=2pt] {$\phi$} (7.0, 1.2);<br />
<br />
\draw (9.5, 0) circle (2);<br />
<br />
\begin{scope}[rotate around={90:(11.5,0)}]<br />
\draw (11.39, 0.25) to[bend left=45] (11.39, -0.25);<br />
\draw (11.5, -0.25) -- (11.5, 0.25);<br />
\draw (11.5, 0.25) -- (11.65, 0.25);<br />
\draw (11.5, -0.25) -- (11.65, -0.25);<br />
<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
The map <math>\phi</math> is:<br />
* Continuous, as it is the composition of continuous maps <math>x\mapsto 2\pi x</math> and <math>t\mapsto e^{it}</math>.<br />
* Injective, because if <math>e^{2\pi i x_1}=e^{2\pi i x_2}</math>, then <math>x_1-x_2\in \mathbb{Z}</math>. Since <math>x_1,x_2\in [0,1)</math>, it follows that <math>x_1=x_2</math>.<br />
* Surjective, since every point of <math>S^1</math> can be written as <math>e^{2\pi i x}</math> for some <math>x\in [0,1)</math>.<br />
Hence <math>\phi</math> is a continuous bijection.<br />
<br />
However, <math>\phi</math> is not a homeomorphism.<br />
<br />
Consider the sequence<math display="block">z_n = e^{2\pi i (1-\tfrac{1}{n})} \in S^1.</math><br />
Then <math>z_n \to 1 = e^{2\pi i \cdot 0}</math> in <math>S^1</math>. But<br />
<math display="block">\phi^{-1}(z_n) = 1-\tfrac{1}{n} \to 1,</math><br />
which does not converge to <math>\phi^{-1}(1)=0</math> in <math>[0,1)</math>. Thus <math>\phi^{-1}</math> is not continuous.<br />
}}<br />
<br />
== Topological invariants ==<br />
A [[topological invariant]] is a property of a topological space that is preserved under homeomorphisms. In other words, if two spaces are homeomorphic, they either both possess the property or both do not. Invariants are the important tools to classify topological spaces. If two spaces differ in any topological invariant, they cannot be homeomorphic. Conversely, showing that two spaces share many invariants is often the first step on proving they are homeomorpic, though it is never sufficient by itself.<br />
<br />
=== Common topological invariants ===<br />
<br />
* [[Connectedness]]<br />
* [[Compactness]]<br />
* [[Hausdorff space|Hausdorff property]]<br />
* [[Cardinality]] of the space<br />
<br />
=== Algebraic invariants ===<br />
More powerful invariants come from [[algebraic topology]], which assigns algebraic objects to topological spaces.<br />
<br />
* [[Fundamental group]]<br />
* [[Homology group]]<br />
* [[Higher homotopy group]]<br />
<br />
==Homeomorphism group==<br />
The collection of all [[autohomeomorphisms]] of a topological space <math>X</math> forms a [[group]] under composition operation, known as the homeomorphism group of <math>X</math>, denoted <math>\operatorname{Homeo}(X)</math>. The homeomorphism group captures the symmetry in topology. It describes the ways in which a topological space can be continuously transformed onto itself.<br />
<br />
The homeomorphism group <math>\operatorname{Homeo}(X)</math> is a faithful [[group action]] on its underlying set <math>X</math>. It moves points in <math>X</math> continuously onto <math>X</math> itself, and the topological structure of <math>X</math> is also reflected in the algebraic invariants such as the [[Orbit|orbits]] and [[Stabilizer|stabilizers]] of the action.<br />
<br />
For example, consider the 2-sphere <math>S^2</math> as a thin rubber membrane tightly wraped around a ball. Each autohomeomorphism of <math>S^2</math>, which is an element in <math>\operatorname{Homeo}(S^2)</math>, corresponds to a continuous deformation of this membrane. This operation can be stretching, bending, twisting, or any composition of these operations, so the rubber always remains attached to the ball.<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=5mm]{standalone}<br />
\usetikzlibrary{arrows.meta, positioning, calc}<br />
\usepackage{amssymb}<br />
\begin{document}<br />
\begin{tikzpicture}[scale=1.2, >=Stealth]<br />
\begin{scope}[shift={(0,0)}]<br />
\shade[ball color=cyan, opacity=0.2] (0,0) circle (2);<br />
<br />
\draw[cyan!60!blue, thick] (-2,0) arc (180:360:2 and 0.25);<br />
\draw[cyan!60!blue, thin, dashed] (2,0) arc (0:180:2 and 0.25);<br />
<br />
\foreach \y in {-1.2, -0.6, 0.6, 1.2} {<br />
\pgfmathsetmacro{\x}{sqrt(4-\y*\y)}<br />
\draw[cyan!70!black, thin] (-\x, \y) arc (180:360:\x cm and 0.25cm);<br />
\draw[cyan!70!black, thin, dashed] (\x, \y) arc (0:180:\x cm and 0.25cm);<br />
}<br />
<br />
\draw[cyan!70!black, thin] (0,2) arc (90:-90:0.6 and 2);<br />
\draw[cyan!70!black, thin] (0,2) arc (90:-90:1.3 and 2);<br />
\draw[cyan!70!black, thin] (0,2) arc (90:270:0.6 and 2);<br />
\draw[cyan!70!black, thin] (0,2) arc (90:270:1.3 and 2);<br />
\draw[cyan!70!black, thin] (0,2) -- (0,-2);<br />
<br />
\filldraw[fill=yellow!60, draw=yellow!60!black, thick, opacity=0.7]<br />
(0.7, 1.1) .. controls (1.0, 1.3) and (1.3, 1.1) .. (1.4, 0.5) <br />
.. controls (1.1, 0.1) and (0.1, 0.6) .. (0.7, 1.1) <br />
-- cycle;<br />
\node[yellow!60!black, font=\bfseries] at (1.1, 0.9) {$U$};<br />
<br />
\filldraw[black] (0.8, 0.9) circle (1pt) node[anchor=south east, scale=0.8] {$x$};<br />
\filldraw[black] (1.1, 0.7) circle (1pt) node[anchor=north west, scale=0.8] {$y$};<br />
<br />
\node[below, text=black] at (0,-2.3) {$X$};<br />
\draw[cyan!60!blue, thick] (0,0) circle (2);<br />
\end{scope}<br />
<br />
\draw[->, thick, black] (2.5, 0) -- (4.5, 0) <br />
node[midway, above, font=\small] {$\phi \in \mathrm{Homeo}(X)$};<br />
<br />
\begin{scope}[shift={(7,0)}]<br />
\shade[ball color=cyan, opacity=0.2] (0,0) circle (2);<br />
<br />
\draw[cyan!60!blue, thick] (-2.0,0.0) .. controls (-0.9, -0.3) and (0.9, 0.3) .. (2.0, 0.0);<br />
\draw[cyan!60!blue, thin, dashed] (2.0,0.0) .. controls (0.9, 0.6) and (-0.9, 0.0) .. (-2.0, 0.0);<br />
<br />
\draw[cyan!70!black, thin] (-1.6, 1.2) .. controls (-0.6, 0.9) and (0.6, 1.5) .. (1.6, 1.2);<br />
\draw[cyan!70!black, thin, dashed] (1.6, 1.2) .. controls (0.6, 1.6) and (-0.6, 1.0) .. (-1.6, 1.2);<br />
\draw[cyan!70!black, thin] (-1.9, 0.6) .. controls (-0.9, 0.3) and (0.9, 0.9) .. (1.9, 0.6);<br />
\draw[cyan!70!black, thin, dashed] (1.9, 0.6) .. controls (0.9, 1.1) and (-0.9, 0.5) .. (-1.9, 0.6);<br />
\draw[cyan!70!black, thin] (-1.9, -0.6) .. controls (-0.9, -0.9) and (0.9, -0.3) .. (1.9, -0.6);<br />
\draw[cyan!70!black, thin, dashed] (1.9, -0.6) .. controls (0.9, -0.1) and (-0.9, -0.5) .. (-1.9, -0.6);<br />
\draw[cyan!70!black, thin] (-1.6, -1.2) .. controls (-0.6, -1.5) and (0.6, -0.9) .. (1.6, -1.2);<br />
\draw[cyan!70!black, thin, dashed] (1.6, -1.2) .. controls (0.6, -0.8) and (-0.6, -1.4) .. (-1.6, -1.2);<br />
<br />
\draw[cyan!70!black, thin] (0, 2) .. controls (-1.9, 0.8) and (-1.7, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin] (0, 2) .. controls (-1.1, 0.8) and (-0.9, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin] (0, 2) .. controls (-0.3, 0.8) and (-0.1, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin] (0, 2) .. controls (0.5, 0.8) and (0.7, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin] (0, 2) .. controls (1.3, 0.8) and (1.5, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin, dashed] (0, 2) .. controls (-1.3, 0.8) and (-0.8, -1.2) .. (0, -2);<br />
\draw[cyan!70!black, thin, dashed] (0, 2) .. controls (0.9, 0.8) and (0.5, -1.2) .. (0, -2);<br />
<br />
\filldraw[fill=yellow!60, draw=yellow!60!black, thick, opacity=0.7]<br />
(0.05, 1.3) <br />
.. controls (0.4, 1.4) and (0.9, 1.5) .. (1.2, 0.7) <br />
.. controls (0.9, 0.3) and (0.4, 0.15) .. (0.05, 0.5) <br />
.. controls (-0.1, 0.8) and (-0.1, 1.1) .. (0.05, 1.3) <br />
-- cycle;<br />
\node[yellow!60!black, font=\bfseries] at (0.6, 0.95) {$\phi(U)$};<br />
<br />
\filldraw[black] (0.4, 1.1) circle (1pt) node[anchor=south east, scale=0.8] {$\phi(x)$};<br />
\filldraw[black] (0.8, 0.6) circle (1pt) node[anchor=north west, scale=0.8] {$\phi(y)$};<br />
<br />
\node[below, text=black] at (0,-2.3) {$\phi\cdot X$};<br />
\draw[cyan!60!blue, thick] (0,0) circle (2);<br />
\end{scope}<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
Under the natural action of <math>\operatorname{Homeo}(S^2)</math>, every point on the sphere can be moved continuously to any other point. This example shows how the homeomorphism group captures the symmetry of a topological space in the perspective of continuity.<br />
<br />
==See also==<br />
* [[Homotopy]]<br />
* [[Topology]]<br />
* [[Homeomorphism group]]<br />
<br />
[[Category:Topopogy]]<br />
<br />
==Terminology==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homeomorphism | homéomorphisme | Homeomorphismus | 同胚 | 同胚 | 同相写像 }}<br />
{{Terminology_table/row | homeomorphic | homéomorphe | homeomorph | 同胚的 | 同胚的 | 同相 }}<br />
{{Terminology_table/row | topological invariant | invariant topologique | topologische Invariante | 拓扑不变量 | 拓撲不變量 | 位相不変量 }}<br />
{{Terminology_table/row | autohomeomorphism | autohoméomorphisme | Selbsthomöomorphismus | 自同胚 | 自同胚 | 自己同相写像 }}<br />
{{Terminology_table/row | homeomorphism group | groupe des homéomorphismes | Homöomorphismengruppe | 同胚群 | 同胚群 | 同相群 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homeomorphism&diff=101Homeomorphism2026-04-11T07:19:04Z<p>InfernalAtom683: </p>
<hr />
<div>[[File:Topology joke.jpg|thumb|250x250px|A homeomorphism that turns a coffee mug into a donut continuously.]]<br />
A '''homeomorphism''' is a special type of [[function]] between two [[Topological space|topological spaces]], that establishes that the two spaces are fundamentally the same from a topological perspective. Specifically, it is a [[Continuous function|continuous]] [[bijective]] function whose [[inverse function]] is also continuous. Homeomorphisms are the [[Isomorphism|isomorphisms]] in the [[category of topological spaces]] <math>\mathsf{Top}</math>, which preserves all [[topological properties]] of a topological space. If such a function exists between two spaces, they are said to be '''homeomorphic'''. <br />
<br />
Intuitively, two spaces are homeomorphic if one can be continuously deformed into the other by stretching, bending, and twisting, without cutting, tearing, or gluing. A typical intuitive example is that a mug with a handle is homeomorphic to a donut. This concept is distinct from [[Homotopy#Homotopy equivalence|homotopy equivalence]], which allows deformations that involve collapsing. For instance, a solid ball can be continuously shrunk to a point by a homotopy, but such a deformation is not a homeomorphism because it is not bijective and the inverse would not be continuous.<br />
<br />
== Definitions ==<br />
A function <math>f</math> between topological spaces <math>X</math> and <math>Y</math> is called a '''homeomorphism''', if:<br />
<br />
* <math>f</math> is continuous,<br />
* <math>f</math> is bijective,<br />
* <math>f^{-1}</math> is continuous.<br />
<br />
Two topological spaces <math>X</math> and <math>Y</math> are called '''homeomorphic''' if there exists a homeomorphism between them, denoted <math>X\cong Y</math>.<br />
<br />
=== Equivalent Definitions ===<br />
A homeomorphism is a bijection that is continuous and [[Open function|open]], or continuous and [[Closed function|closed]].<br />
<br />
== Properties ==<br />
{{Property|property=The composition of two homeomorphisms is again a homeomorphism.}}{{Proof|proof=Let <math>f\colon X \to Y</math> and <math>g\colon Y \to Z</math> be homeomorphisms. Then:<br />
<br />
* <math>g \circ f\colon X \to Z</math> is bijective, since the composition of two bijections is a bijection.<br />
<br />
* <math>g \circ f</math> is continuous, as the composition of two continuous functions.<br />
<br />
* The inverse is <math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}</math>, which is continuous because it is the composition of the continuous functions <math>g^{-1}</math> and <math>f^{-1}</math>.<br />
<br />
Thus <math>g \circ f</math> satisfies all requirements of a homeomorphism.}}<br />
<br />
<br />
{{Property|property=The inverse of a homeomorphism is again a homeomorphism.}}<br />
{{Proof|proof=Let <math>f\colon X \to Y</math> be a homeomorphism. Then:<br />
* <math>f^{-1}</math> is continuous by definition,<br />
* <math>f^{-1}</math> is bijective, since the inverse of a bijection is again a bijection,<br />
* <math>\left(f^{-1}\right)^{-1}=f</math> is continuous by definition.}}<br />
<br />
<br />
{{Property|property=Homeomorphism is an [[equivalence relation]].}}<br />
<br />
{{Proof|proof=<br />
* '''Reflexivity''': The identity map <math>\operatorname{id}_X\colon X\to X</math> is a continuous bijection on any topological space <math>X</math>, whose inverse is itself. Thus <math>\operatorname{id}_X</math> is a homeomorphism.<br />
* '''Symmetry''': If <math>f\colon X\to Y</math> is a homeomorphism, then its inverse <math>f^{-1}\colon Y\to X</math> is again a homeomorphism.<br />
* '''Transitivity''': If <math>f\colon X\to Y</math> and <math>g\colon Y\to Z</math> are homeomorphisms, then <math>g\circ f: X\to Z</math> is again a homeomorphism.<br />
}}<br />
<br />
== Examples ==<br />
<br />
=== Open interval ===<br />
The [[open interval]] <math>(0,1)</math> is homeomorphic to <math>\mathbb{R}</math>.<br />
{{Proof|proof=The map <math>f\colon (0,1)\to \mathbb{R}</math> defined by<br />
<math display="block">f(x)=\tan\left(\pi\left(x-\dfrac12\right)\right)</math><br />
is a homeomorphism. Indeed, <math>f</math> is continuous because it is a composition of continuous functions. The restriction <math>\tan\colon (-\pi/2,\pi/2)\to\mathbb{R}</math> is bijective with continuous inverse <math>\arctan\colon \mathbb{R}\to(-\pi/2,\pi/2)</math>. Therefore <math>f</math> is bijective and its inverse<br />
<math display="block">f^{-1}(y)=\dfrac1\pi\arctan(y)+\dfrac12</math><br />
is continuous. Thus <math>f</math> is a homeomorphism.}}<br />
<br />
=== Stereographic projection ===<br />
The [[Euclidean plane]] <math>\mathbb{R}^2</math> is homeomorphic to the [[2-sphere]] minus one point, denoted <math>S^2 \setminus \{N\}</math> where <math>N=(0,0,1)</math> is the [[north pole]].<br />
<br />
{{Proof|proof=<br />
<div style="float:right; width:320px; margin:0 0 0.5em 1em;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=15pt]{standalone}<br />
\usepackage{tikz-3dplot}<br />
\usetikzlibrary{calc, arrows.meta}<br />
\begin{document}<br />
\def\viewTheta{70}<br />
\def\viewPhi{20}<br />
\tdplotsetmaincoords{\viewTheta}{\viewPhi}<br />
<br />
\begin{tikzpicture}[tdplot_main_coords, scale=2, line cap=round, line join=round]<br />
<br />
\def\R{1}<br />
\coordinate (O) at (0,0,0);<br />
\coordinate (N) at (0,0,\R);<br />
<br />
\def\thetaS{60}<br />
\def\phiS{30}<br />
\pgfmathsetmacro{\px}{\R * sin(\thetaS) * cos(\phiS)}<br />
\pgfmathsetmacro{\py}{\R * sin(\thetaS) * sin(\phiS)}<br />
\pgfmathsetmacro{\pz}{\R * cos(\thetaS)}<br />
\coordinate (P) at (\px, \py, \pz);<br />
<br />
\pgfmathsetmacro{\ux}{\px / (1 - \pz)}<br />
\pgfmathsetmacro{\uy}{\py / (1 - \pz)}<br />
\coordinate (Pprime) at (\ux, \uy, 0);<br />
<br />
\pgfmathsetmacro{\eqFrontStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\eqBackStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqBackEnd}{\viewPhi+180}<br />
<br />
\pgfmathsetmacro{\cotViewTheta}{cos(\viewTheta)/sin(\viewTheta)}<br />
\pgfmathsetmacro{\cotThetaS}{cos(\thetaS)/sin(\thetaS)}<br />
\pgfmathsetmacro{\cosAlpha}{max(min(-\cotThetaS * \cotViewTheta, 1), -1)}<br />
\pgfmathsetmacro{\alpha}{acos(\cosAlpha)}<br />
<br />
\pgfmathsetmacro{\latFrontStart}{\viewPhi-180}<br />
\pgfmathsetmacro{\latFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\latBackStart}{\viewPhi}<br />
\pgfmathsetmacro{\latBackEnd}{\viewPhi+180}<br />
<br />
\draw[thick, black] (-1.2,0,0) -- (0,0,0);<br />
\draw[thick, black] (0,-3,0) -- (0,0,0);<br />
\draw[thick, black] (0,0,-1.2) -- (0,0,0);<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (2.2,0,0) node[anchor=north east]{$x$};<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (0,3.0,0) node[anchor=north west]{$y$};<br />
\draw[thick, dashed] (0,0,0) -- (N);<br />
\draw[thick, ->, >=Stealth] (N) -- (0,0,1.8) node[anchor=south]{$z$};<br />
\begin{scope}[tdplot_screen_coords]<br />
\shade[ball color=cyan, opacity=0.15] (0,0) circle (\R);<br />
\draw[cyan!60!blue, thick] (0,0) circle (\R);<br />
\end{scope}<br />
<br />
\tdplotdrawarc[cyan!60!blue, thick]{(O)}{\R}{\eqFrontStart}{\eqFrontEnd}{}{}<br />
\tdplotdrawarc[cyan!60!blue, thin, dashed]{(O)}{\R}{\eqBackStart}{\eqBackEnd}{}{}<br />
<br />
\pgfmathsetmacro{\rLat}{\R * sin(\thetaS)}<br />
\coordinate (CenterLat) at (0,0,\pz);<br />
<br />
\tdplotsetrotatedcoords{0}{0}{0}<br />
\tdplotsetrotatedcoordsorigin{(CenterLat)}<br />
\tdplotdrawarc[cyan!70!black, thin, dashed, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latBackStart}{\latBackEnd}{}{}<br />
\tdplotdrawarc[cyan!70!black, thin, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latFrontStart}{\latFrontEnd}{}{}<br />
\tdplotsetrotatedcoordsorigin{(O)}<br />
<br />
\draw[red, thick, dashed] (N) -- (P);<br />
\draw[red, thick, ->, >=Stealth] (P) -- (Pprime);<br />
<br />
\fill[black] (N) circle (0.8pt) node[anchor=south east] {$N$};<br />
\fill[red] (P) circle (1pt) node[anchor=south west, text=black] {$(x,y,z)$};<br />
\fill[red] (Pprime) circle (1pt) node[anchor=north west, text=black] {$p(x,y,z) = (u,v)$};<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
Define the [[stereographic projection]] <math>p\colon S^2 \setminus \{N\} \to \mathbb{R}^2</math> by<br />
<math display="block">p(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right).</math><br />
This map is continuous because it is a rational function with denominator nonzero (since <math>z<1</math> on <math>S^2\setminus\{N\}</math>).<br />
<br />
The inverse map <math>p^{-1}\colon \mathbb{R}^2 \to S^2 \setminus \{N\}</math> is given by<br />
<math display="block">p^{-1}(u,v) = \left( \frac{2u}{u^2+v^2+1}, \frac{2v}{u^2+v^2+1}, \frac{u^2+v^2-1}{u^2+v^2+1} \right).</math><br />
This is also continuous as a composition of continuous functions. One verifies that <math>p \circ p^{-1} = \text{id}_{\mathbb{R}^2}</math> and <math>p^{-1} \circ p = \operatorname{id}_{S^2\setminus\{N\}}</math> by direct substitution. Hence <math>p</math> is a homeomorphism.<br />
}}<br />
<br />
=== Quotient space ===<br />
The unit interval <math>[0,1]</math> with the endpoints identified (the quotient space <math>[0,1]/\sim</math> where <math>0\sim 1</math>) is homeomorphic to the circle <math>S^1</math>.<br />
<br />
{{Proof|proof=Define the map <math>f\colon [0,1] \to S^1</math> by <math display="block">f(t)=(\cos(2\pi t), \sin(2\pi t)).</math> This map is continuous and [[Surjection|surjective]], and satisfies <math>f(0)=f(1)=(1,0)</math>.<br />
<br />
Consider the equivalence relation <math>\sim</math>, and let <math>q\colon [0,1]\to [0,1]/\sim</math> be the [[quotient map]]. By the [[universal property]] of the quotient map, there exists a unique continuous map <math>\tilde{f}\colon [0,1]/\sim \to S^1</math> such that <math>\tilde{f} \circ q = f</math>; that is, the following diagram commutes.<br />
<div style="text-align: center;"><br />
<br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
{[0,1]} && {S^1} \\<br />
& {[0,1]/{\sim}} \arrow["f", from=1-1, to=1-3]<br />
\arrow["q"', from=1-1, to=2-2]<br />
\arrow["{\exists! \tilde{f}}"', dashed, from=2-2, to=1-3]<br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
The map <math>\tilde{f}</math> is bijective because:<br />
* Surjectivity follows from surjectivity of <math>f</math>;<br />
* [[Injection|Injectivity]] holds because <math display="block">\tilde{f}([t])=\tilde{f}([s])\Rightarrow t=s \text{ or } \{t,s\}=\{0,1\},</math> but in the latter case <math>[t]=[s]</math> in the quotient. <br />
<br />
Hence <math>\tilde{f}</math> is a continuous bijection.<br />
<br />
The space <math>[0,1]/\sim</math> is compact as the quotient of a [[compact space]], and <math>S^1</math> is [[Hausdorff space|Hausdorff]]. By the [[Compact-to-Hausdorff theorem]], a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.<br />
<br />
Therefore <math>\tilde{f}</math> is a homeomorphism.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz">\documentclass[tikz,border=10pt]{standalone}<br />
\usepackage{amsmath}<br />
\usetikzlibrary{arrows.meta,calc}<br />
\begin{document}<br />
<br />
\begin{tikzpicture}[<br />
>={Stealth[scale=1.1]},<br />
dot/.style={circle,fill=black,inner sep=1.6pt},<br />
label text/.style={font=\Large,align=center}<br />
]<br />
<br />
\def\r{1.4}<br />
\def\gap{50}<br />
<br />
\coordinate (C1) at (0,0);<br />
\coordinate (C2) at (5.5,0);<br />
\coordinate (C3) at (11,0);<br />
<br />
\begin{scope}[shift={(C1)}]<br />
\draw[thick] (-\r,0) coordinate (A) -- (\r,0) coordinate (B);<br />
\node[dot] at (A) {};<br />
\node[dot] at (B) {};<br />
\node[label text] at (0,-2.6) {$[0,1]$};<br />
\coordinate (R1) at (\r,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C1)+(2,0)$) -- ($(C2)+(-2,0)$)<br />
node[midway,above] {$q$};<br />
<br />
\begin{scope}[shift={(C2)}]<br />
\draw[thick]<br />
(180-\gap:\r)<br />
arc[start angle=180-\gap,end angle=360+\gap,radius=\r];<br />
<br />
\node[dot] (L) at (180-\gap:\r) {};<br />
\node[dot] (R) at (\gap:\r) {};<br />
<br />
\draw[dashed,->,thick]<br />
(L) .. controls +(0,0) and +(-0.8,-0.1) .. (90:\r);<br />
\draw[dashed,->,thick]<br />
(R) .. controls +(0,0) and +(0.8,-0.1) .. (90:\r);<br />
<br />
\node[label text] at (0,-2.6)<br />
{$[0,1]/\sim$ \\[-0.4ex]\normalsize $(0\sim1)$};<br />
<br />
\coordinate (R2) at (2,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C2)+(2,0)$) -- ($(C3)+(-2,0)$)<br />
node[midway,above] {$\overset{\tilde{f}}{\cong}$};<br />
<br />
\begin{scope}[shift={(C3)}]<br />
\draw[thick] (0,0) circle (\r);<br />
\node[dot] at (90:\r) {};<br />
\node[label text] at (0,-2.6) {$S^{1}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}</kroki><br />
</div><br />
<br />
}}<br />
<br />
== Counterexamples ==<br />
<math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>.<br />
<br />
{{Proof|proof=For contradiction, suppose that there exists a homeomorphism <math>f\colon \mathbb{R}\to\mathbb{R}^2</math>.<br />
<br />
Consider the subspace <math>\mathbb{R}\setminus\{0\}</math> of <math>\mathbb{R}</math>. The [[restriction]] on it, <math>\left.f\right|_{\mathbb{R}\setminus\{0\}}\colon \mathbb{R}\setminus\{0\}\to \mathbb{R}^2\setminus\{f(0)\}</math> is also a homeomorphism.<br />
<br />
However, <math>\mathbb{R}\setminus\{0\}</math> has two connected components, <math>(-\infty,0)</math> and <math>(0,\infty)</math>, while <math>\mathbb{R}^2\setminus\{f(0)\}</math> is connected, which contradicts the assumption that the two spaces are homeomorphic.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz, border=10pt]{standalone}<br />
\usepackage{amsmath, amssymb}<br />
<br />
\begin{document}<br />
\begin{tikzpicture}<br />
\begin{scope}[xshift=-5cm]<br />
\draw[thick] (-3, 0) -- (3, 0);<br />
<br />
\filldraw[fill=white, draw=black, thick] (0, 0) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2.5, 0) -- (-1, 0);<br />
\fill[cyan!60!blue] (-2.5, 0) circle (2pt);<br />
\fill[cyan!60!blue] (-1, 0) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (1.3, 0) -- (2.2, 0);<br />
\fill[cyan!60!blue] (1.3, 0) circle (2pt);<br />
\fill[cyan!60!blue] (2.2, 0) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -2) {$\mathbb{R} \setminus \{0\}$};<br />
\end{scope}<br />
<br />
\begin{scope}[xshift=4cm]<br />
\draw[thick] (-3.5, -2.5) -- (3.5, -2.5) -- (3.5, 2.5) -- (-3.5, 2.5) -- cycle;<br />
<br />
\filldraw[fill=white, draw=black, thick] (-0.3, -0.2) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2, 1.5) .. controls (-0.5, 1) and (-0.8, -1) .. (-1.2, -1.8);<br />
\fill[cyan!60!blue] (-2, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (-1.2, -1.8) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 1.5) .. controls (1.5, 1.8) and (2.5, 1) .. (2, 0.5);<br />
\fill[cyan!60!blue] (-0.3, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (2, 0.5) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 0.5) .. controls (1, 0) and (1, -1) .. (2.5, -2);<br />
\fill[cyan!60!blue] (-0.3, 0.5) circle (2pt);<br />
\fill[cyan!60!blue] (2.5, -2) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.5, -0.8) .. controls (0, -1.8) and (1, -1.5) .. (0.8, -1);<br />
\fill[cyan!60!blue] (-0.5, -0.8) circle (2pt);<br />
\fill[cyan!60!blue] (0.8, -1) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -3.5) {$\mathbb{R}^2 \setminus \{f(0)\}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
Hence, no such homeomorphism exists; therefore <math>\mathbb{R}</math> is not homeomorphic to <math>\mathbb{R}^2</math>}}The map from the interval <math>[0,1)</math> to the 1-sphere <math>S^1</math>,<br />
<math display="block">\phi\colon [0,1)\to S^1,\quad x\mapsto e^{2\pi ix}</math><br />
is continuous and bijective, but not a homeomorphism. <br />
{{Proof|proof=<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass{article}<br />
\usepackage{tikz}<br />
\usetikzlibrary{arrows.meta}<br />
<br />
\begin{document}<br />
<br />
\begin{tikzpicture}<br />
\draw (0,0) -- (4,0);<br />
<br />
\draw (0.15, 0.25) -- (0, 0.25) -- (0, -0.25) -- (0.15, -0.25);<br />
<br />
\draw (3.9, 0.25) to[bend left=45] (3.9, -0.25);<br />
<br />
\draw[-{Stealth[length=3mm, width=2mm]}] (4.5, 1.2) to[bend left=30] node[above=2pt] {$\phi$} (7.0, 1.2);<br />
<br />
\draw (9.5, 0) circle (2);<br />
<br />
\begin{scope}[rotate around={90:(11.5,0)}]<br />
\draw (11.39, 0.25) to[bend left=45] (11.39, -0.25);<br />
\draw (11.5, -0.25) -- (11.5, 0.25);<br />
\draw (11.5, 0.25) -- (11.65, 0.25);<br />
\draw (11.5, -0.25) -- (11.65, -0.25);<br />
<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
The map <math>\phi</math> is:<br />
* Continuous, as it is the composition of continuous maps <math>x\mapsto 2\pi x</math> and <math>t\mapsto e^{it}</math>.<br />
* Injective, because if <math>e^{2\pi i x_1}=e^{2\pi i x_2}</math>, then <math>x_1-x_2\in \mathbb{Z}</math>. Since <math>x_1,x_2\in [0,1)</math>, it follows that <math>x_1=x_2</math>.<br />
* Surjective, since every point of <math>S^1</math> can be written as <math>e^{2\pi i x}</math> for some <math>x\in [0,1)</math>.<br />
Hence <math>\phi</math> is a continuous bijection.<br />
<br />
However, <math>\phi</math> is not a homeomorphism.<br />
<br />
Consider the sequence<math display="block">z_n = e^{2\pi i (1-\tfrac{1}{n})} \in S^1.</math><br />
Then <math>z_n \to 1 = e^{2\pi i \cdot 0}</math> in <math>S^1</math>. But<br />
<math display="block">\phi^{-1}(z_n) = 1-\tfrac{1}{n} \to 1,</math><br />
which does not converge to <math>\phi^{-1}(1)=0</math> in <math>[0,1)</math>. Thus <math>\phi^{-1}</math> is not continuous.<br />
}}<br />
== Topological invariants ==<br />
A [[topological invariant]] is a property of a topological space that is preserved under homeomorphisms. In other words, if two spaces are homeomorphic, they either both possess the property or both do not. Invariants are the important tools to classify topological spaces. If two spaces differ in any topological invariant, they cannot be homeomorphic. Conversely, showing that two spaces share many invariants is often the first step on proving they are homeomorpic, though it is never sufficient by itself.<br />
<br />
=== Common topological invariants ===<br />
<br />
* [[Connectedness]]<br />
* [[Compactness]]<br />
* [[Hausdorff space|Hausdorff property]]<br />
* [[Cardinality]] of the space<br />
<br />
=== Algebraic invariants ===<br />
More powerful invariants come from [[algebraic topology]], which assigns algebraic objects to topological spaces.<br />
<br />
* [[Fundamental group]]<br />
* [[Homology group]]<br />
* [[Higher homotopy group]]<br />
<br />
==Homeomorphism group==<br />
The collection of all [[autohomeomorphisms]] of a topological space <math>X</math> forms a [[group]] under composition operation, known as the homeomorphism group of <math>X</math>, denoted <math>\operatorname{Homeo}(X)</math>. The homeomorphism group captures the symmetry in topology. It describes the ways in which a topological space can be continuously transformed onto itself.<br />
<br />
The homeomorphism group <math>\operatorname{Homeo}(X)</math> is a faithful [[group action]] on its underlying set <math>X</math>. It moves points in <math>X</math> continuously onto <math>X</math> itself, and the topological structure of <math>X</math> is also reflected in the algebraic invariants such as the [[Orbit|orbits]] and [[Stabilizer|stabilizers]] of the action.<br />
<br />
For example, consider the 2-sphere <math>S^2</math> as a thin rubber membrane tightly wraped around a ball. Each autohomeomorphism of <math>S^2</math>, which is an element in <math>\operatorname{Homeo}(S^2)</math>, corresponds to a continuous deformation of this membrane. This operation can be stretching, bending, twisting, or any composition of these operations, so the rubber always remains attached to the ball.<br />
<div style="text-align: center;"><br />
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(0.7, 1.1) .. controls (1.0, 1.3) and (1.3, 1.1) .. (1.4, 0.5) <br />
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<br />
\filldraw[fill=yellow!60, draw=yellow!60!black, thick, opacity=0.7]<br />
(0.05, 1.3) <br />
.. controls (0.4, 1.4) and (0.9, 1.5) .. (1.2, 0.7) <br />
.. controls (0.9, 0.3) and (0.4, 0.15) .. (0.05, 0.5) <br />
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<br />
\node[below, text=black] at (0,-2.3) {$\phi\cdot X$};<br />
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</kroki><br />
</div><br />
Under the natural action of <math>\operatorname{Homeo}(S^2)</math>, every point on the sphere can be moved continuously to any other point. This example shows how the homeomorphism group captures the symmetry of a topological space in the perspective of continuity.<br />
<br />
==See also==<br />
* [[Homotopy]]<br />
* [[Topology]]<br />
* [[Homeomorphism group]]<br />
<br />
[[Category:Topopogy]]<br />
<br />
==Terminology==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homeomorphism | homéomorphisme | Homeomorphismus | 同胚 | 同胚 | 同相写像 }}<br />
{{Terminology_table/row | homeomorphic | homéomorphe | homeomorph | 同胚的 | 同胚的 | 同相 }}<br />
{{Terminology_table/row | topological invariant | invariant topologique | topologische Invariante | 拓扑不变量 | 拓撲不變量 | 位相不変量 }}<br />
{{Terminology_table/row | autohomeomorphism | autohoméomorphisme | Selbsthomöomorphismus | 自同胚 | 自同胚 | 自己同相写像 }}<br />
{{Terminology_table/row | homeomorphism group | groupe des homéomorphismes | Homöomorphismengruppe | 同胚群 | 同胚群 | 同相群 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Hausdorff_space&diff=100Hausdorff space2026-04-11T07:16:30Z<p>InfernalAtom683: </p>
<hr />
<div>A '''Hausdorff space''' (or '''<math>T_2</math> space''') is a type of [[topological space]] in which points can be "cleanly separated" by neighborhoods. Specifically, for any two distinct points, there exist disjoint [[Open set|open sets]] containing each point. Consequently, Hausdorff property ensures that limits of sequences are unique when they exist.<br />
<br />
== Definitions ==<br />
A topological space <math>(X,\tau)</math> is Hausdorff, if for any two points <math>x,y\in X</math>, there exists two disjoint open sets <math>U,V\in \tau</math>, <math>U\cap V=\varnothing</math>, such that <math>x\in U</math> and <math>y\in V</math>.<br />
<br />
=== Equivalent Definitions ===<br />
Any convergent [[sequence]] in <math>X</math> has at most one limit.<br />
<br />
== Properties ==<br />
{{Property|property=Subspaces of Hausdorff spaces are Hausdorff.}}{{Proof|proof=Let <math>Y \subseteq X</math> with <math>X</math> Hausdorff. For <math>y_1, y_2 \in Y, y_1 \neq y_2</math>, there exist disjoint open sets <math>U, V \subseteq X</math> with <math>y_1 \in U</math> and <math>y_2 \in V</math>. Then <math>U \cap Y</math> and <math>V \cap Y</math> are disjoint open sets in <math>Y</math> containing <math>y_1</math> and <math>y_2</math>.}}{{Property|property=Finite products of Hausdorff spaces are Hausdorff.}}{{Proof|proof=Let <math>X_1, \dots, X_n</math> be Hausdorff. Consider points <math>(x_1, \dots, x_n) \neq (y_1, \dots, y_n)</math>. There exists an index <math>i</math> with <math>x_i \neq y_i</math>.<br />
<br />
Since <math>X_i</math> is Hausdorff, choose disjoint open sets <math>U_i, V_i \subseteq X_i</math> containing <math>x_i</math> and <math>y_i</math>. Then <math>U_1 \times \dots \times U_n</math> and <math>V_1 \times \dots \times V_n</math> are disjoint open sets containing the two points.}}{{Property|property=Compact subsets of Hausdorff spaces are closed.}}{{Proof|proof=Let <math>K \subseteq X</math> be compact and <math>X</math> Hausdorff. For any <math>x \in X \setminus K</math>, for each <math>y \in K</math> choose disjoint open sets <math>U_y\ni x</math> and <math>V_y \ni y</math>. <br />
<br />
The collection <math>\{V_y \mid y \in K\}</math> covers <math>K</math>. By compactness, finitely many <math>V_{y_1}, \dots, V_{y_n}</math> cover <math>K</math>. Then <math display="block">U = \bigcap_{i=1}^n U_{y_i}</math> is open, contains <math>x</math>, and is disjoint from <math>K</math>. Hence <math>K</math> is closed.}}<br />
<br />
== Examples ==<br />
Every metric space is a Hausdorff space.<br />
{{Proof|proof=<div style="float:right; width:320px; margin:0 0 0.5em 1em;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=8pt]{standalone}<br />
\usetikzlibrary{calc}<br />
\usepackage{amsmath}<br />
\begin{document}<br />
\begin{tikzpicture}[scale=1.3]<br />
<br />
\def\r{1.5}<br />
<br />
\coordinate (x) at (0,0);<br />
<br />
\def\ang{20}<br />
<br />
\coordinate (y) at ({2*\r*cos(\ang)}, {2*\r*sin(\ang)});<br />
<br />
\draw[fill=blue!25!cyan, opacity=0.2, dashed, thick] (x) circle (\r);<br />
\draw[fill=blue!25!cyan, opacity=0.2, dashed, thick] (y) circle (\r);<br />
<br />
\fill (x) circle (2pt);<br />
\fill (y) circle (2pt);<br />
<br />
\node[below left] at (x) {$x$};<br />
\node[above right] at (y) {$y$};<br />
<br />
\draw[black, dashed, thick]<br />
(x) -- ({\r*cos(\ang)}, {\r*sin(\ang)})<br />
node[midway, above, sloped] {\footnotesize $r=\dfrac{d(x,y)}{2}$};<br />
<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki></div><br />
<br />
Let <math>(X,d)</math> be a metric space, take two distinct points <math>x,y</math> such that <math>d(x,y)>0</math>.<br />
<br />
Consider open balls <math>B(x,r)</math> and <math>B(y,r)</math>where <math>r=\frac{d(x,y)}{2}>0</math>. The open balls are both open with <math>x\in B(x,r)</math> and <math>y\in B(y,r)</math>.<br />
<br />
For contradiction, assume there exists <math>z\in B(x,r)\cap B(y,r)</math>, then <math>d(x,z)<r</math> and <math>d(y,z)<r</math>. By triangular inequality, <math display="block">d(x,y)\leq d(x,y)+d(x,z)<r+r=d(x,y),</math> there exists a contradiction.<br />
<br />
Thus, <math>B(x,r)\cap B(y,r)=\varnothing</math>; therefore <math>(X,d)</math> is Hausdorff.}}<br />
<br />
== See also ==<br />
<br />
* [[Topological space]]<br />
* [[Convergence]]<br />
<br />
==Terminology==<br />
{{Terminology_table|<br />
{{Terminology_table/row | Hausdorff space | espace de Hausdorff (espace séparé) | hausdorff-Raum (hausdorffscher Raum) | Hausdorff 空间 | Hausdorff 空間 | ハウスドルフ空間 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homotopy&diff=99Homotopy2026-04-11T07:15:46Z<p>InfernalAtom683: </p>
<hr />
<div>A '''homotopy''' is a continuous deformation between two [[Continuous function|continuous functions]] from one [[topological space]] to another. Specifically, a homotopy between two functions is a continuous map that, for each point in the domain, provides a path from its image under the first function to its image under the second. If such a function exists between two functions, they are said to be homotopic.<br />
<br />
Intuitively, a homotopy is the continuous transformation of paths that varies over time. It shows how one function can be smoothly bent, stretched, or deformed into the other without tearing or folding.<br />
<br />
== Definition ==<br />
If <math>f,g\colon X\to Y</math> are continuous functions between topological spaces <math>X</math> and <math>Y</math>, a homotopy <math>H</math> from <math>f</math> to <math>g</math> is a continuous map<br />
<br />
<math display="block">H\colon[0,1] \times X \to Y</math><br />
<br />
such that for all <math>x \in X</math>:<br />
<br />
* <math>H_0(x) = f(x)</math>,<br />
<br />
* <math>H_1(x) = g(x)</math>.<br />
<br />
Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math><ref group="Note">Not to be confused with "<math>\cong</math>" for homeomorphism.</ref>.<br />
<br />
== Homotopy equivalence ==<br />
Two topological spaces <math>X</math> and <math>Y</math> are '''homotopy equivalent''' if there exist continuous maps<br />
<br />
<math display="block">f\colon X \to Y \quad \text{and} \quad g\colon Y \to X</math><br />
<br />
such that<br />
<br />
* <math>g \circ f \simeq \operatorname{id}_X</math>,<br />
<br />
* <math>f \circ g \simeq \operatorname{id}_Y</math>.<br />
<br />
In this case, <math>X</math> and <math>Y</math> have the same "essential shape" from the perspective of homotopy. This concept is distinct from [[homeomorphism]], which is a stricter condition requiring the maps to be inverses of each other. For instance, a solid [[sphere]] is homotopy equivalent to a single point, but they are not homeomorphic.<br />
<br />
== Note ==<br />
<references group="Note" /><br />
<br />
== See also ==<br />
<br />
* [[Homeomorphism]]<br />
* [[Continuous function]]<br />
* [[Homeotopy]]<br />
* [[Homotopy type theory]]<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homotopy | homotopie | Homotopie | 同伦 | 同倫 | ホモトピー }}<br />
{{Terminology_table/row | homotopic | homotope | homotop | 同伦的 | 同倫的 | ホモトピック }}<br />
{{Terminology_table/row | homotopy equivalence | équivalence d'homotopie | Homotopieäquivalenz | 同伦等价 | 同倫等價 | ホモトピー同値 }}<br />
{{Terminology_table/row | homotopy equivalent | homotopiquement équivalent | homotopieäquivalent | 同伦等价的 | 同倫等價的 | ホモトピー同値な }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homotopy&diff=98Homotopy2026-04-11T07:15:12Z<p>InfernalAtom683: </p>
<hr />
<div>A '''homotopy''' is a continuous deformation between two [[Continuous function|continuous functions]] from one [[topological space]] to another. Specifically, a homotopy between two functions is a continuous map that, for each point in the domain, provides a path from its image under the first function to its image under the second. If such a function exists between two functions, they are said to be homotopic.<br />
<br />
Intuitively, a homotopy is the continuous transformation of paths that varies over time. It shows how one function can be smoothly bent, stretched, or deformed into the other without tearing or folding.<br />
<br />
== Definition ==<br />
If <math>f,g\colonX\to Y</math> are continuous functions between topological spaces <math>X</math> and <math>Y</math>, a homotopy <math>H</math> from <math>f</math> to <math>g</math> is a continuous map<br />
<br />
<math display="block">H\colon[0,1] \times X \to Y</math><br />
<br />
such that for all <math>x \in X</math>:<br />
<br />
* <math>H_0(x) = f(x)</math>,<br />
<br />
* <math>H_1(x) = g(x)</math>.<br />
<br />
Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math><ref group="Note">Not to be confused with "<math>\cong</math>" for homeomorphism.</ref>.<br />
<br />
== Homotopy equivalence ==<br />
Two topological spaces <math>X</math> and <math>Y</math> are '''homotopy equivalent''' if there exist continuous maps<br />
<br />
<math display="block">f\colonX \to Y \quad \text{and} \quad g\colonY \to X</math><br />
<br />
such that<br />
<br />
* <math>g \circ f \simeq \operatorname{id}_X</math>,<br />
<br />
* <math>f \circ g \simeq \operatorname{id}_Y</math>.<br />
<br />
In this case, <math>X</math> and <math>Y</math> have the same "essential shape" from the perspective of homotopy. This concept is distinct from [[homeomorphism]], which is a stricter condition requiring the maps to be inverses of each other. For instance, a solid [[sphere]] is homotopy equivalent to a single point, but they are not homeomorphic.<br />
<br />
== Note ==<br />
<references group="Note" /><br />
<br />
== See also ==<br />
<br />
* [[Homeomorphism]]<br />
* [[Continuous function]]<br />
* [[Homeotopy]]<br />
* [[Homotopy type theory]]<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homotopy | homotopie | Homotopie | 同伦 | 同倫 | ホモトピー }}<br />
{{Terminology_table/row | homotopic | homotope | homotop | 同伦的 | 同倫的 | ホモトピック }}<br />
{{Terminology_table/row | homotopy equivalence | équivalence d'homotopie | Homotopieäquivalenz | 同伦等价 | 同倫等價 | ホモトピー同値 }}<br />
{{Terminology_table/row | homotopy equivalent | homotopiquement équivalent | homotopieäquivalent | 同伦等价的 | 同倫等價的 | ホモトピー同値な }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Category_of_topological_spaces&diff=97Category of topological spaces2026-04-10T14:58:34Z<p>InfernalAtom683: </p>
<hr />
<div>The '''category of topological spaces''', denoted <math>\mathsf{Top}</math> or <math>\mathbf{Top}</math>, is the [[category]] whose objects are [[Topological space|topological spaces]] and whose [[Morphism|morphisms]] are [[Continuous function|continuous functions]].<br />
<br />
== Definition ==<br />
The category <math>\mathsf{Top}</math> consists of:<br />
<br />
* <math>\operatorname{ob}(\mathsf{Top})</math> is the class of all topological spaces;<br />
* <math>\operatorname{mor}(\mathsf{Top})</math> is the class of all continuous functions between topological spaces;<br />
* The composition operation <math>\circ</math> is given by the composition of ordinary functions.<br />
<br />
== Subcategories ==<br />
Several important subcategories of <math>\mathsf{Top}</math> arise by restricting objects:<br />
<br />
* <math>\mathsf{Haus}</math>, [[category of Hausdorff spaces]];<br />
* <math>\mathsf{Comp}</math>, [[category of Compact spaces]];<br />
<br />
* <math>\mathsf{Top}_{*}</math>, [[category of pointed topological spaces]];<br />
* <math>\mathsf{CW}</math>, [[category of CW complexes]] or cell complexes.<br />
<br />
== See also ==<br />
<br />
* [[Category theory]]<br />
* [[Morphism]]<br />
* [[Category of sets]]<br />
* [[Category of groups]]<br />
<br />
== Terminology ==<br />
<br />
{{Terminology_table|<br />
{{Terminology_table/row | category of topological spaces | catégorie des espaces topologiques | Kategorie der topologischen Räume | 拓扑空间范畴 | 拓撲空間範疇 | 位相空間の圏 }}<br />
{{Terminology_table/row | topological space | espace topologique | topologischer Raum | 拓扑空间 | 拓撲空間 | 位相空間 }}<br />
{{Terminology_table/row | continuous function | fonction continue | stetige Funktion | 连续函数 | 連續函數 | 連続関数 }}<br />
{{Terminology_table/row | category of Hausdorff spaces | catégorie des espaces de Hausdorff | Kategorie der Hausdorff‑Räume | Hausdorff 空间范畴 | Hausdorff 空間範疇 | ハウスドルフ空間の圏 }}<br />
{{Terminology_table/row | category of compact spaces | catégorie des espaces compacts | Kategorie der kompakten Räume | 紧空间范畴 | 緊空間範疇 | コンパクト空間の圏 }}<br />
{{Terminology_table/row | category of pointed topological spaces | catégorie des espaces pointés | Kategorie der punktierten topologischen Räume | 带点拓扑空间范畴 | 帶點拓撲空間範疇 | 点付き位相空間の圏 }}<br />
{{Terminology_table/row | category of CW complexes | catégorie des complexes CW | Kategorie der CW‑Komplexe | CW复形范畴 | CW複形範疇 | CW複体の圏 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=First_isomorphism_theorem&diff=96First isomorphism theorem2026-04-10T13:34:32Z<p>InfernalAtom683: </p>
<hr />
<div>The '''first isomorphism theorem''' is a fundamental result in [[abstract algebra]] that describes the relationship between a [[homomorphism]], its [[Kernel (algebra)|kernel]], and its [[Image (mathematics)|image]]. The theorem appears uniformly across algebraic structures such as [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], and [[Module (mathematics)|modules]], and serves as a prototype for many structural results in algebra. Specifically, given a homeomorphism, the quotient of its domain by its kernel is isomorphic to its image.<br />
<br />
== Group theory ==<br />
<br />
=== Statement ===<br />
Let <math>G</math> and <math>H</math> be [[Group|groups]] and <math>f\colon g\to H</math> a group homomorphism. Then,<br />
<br />
# The kernel of <math>f</math>, <math>\ker f\trianglelefteq G</math> is a normal subgroup of <math>G</math>.<br />
# The image of <math>f</math>, <math>\operatorname{im}f<G</math> is a subgroup of <math>H</math>.<br />
# <math>G/\ker f \cong \operatorname{im} f</math>.<br />
<br />
=== Proof ===<br />
{{Proof|title=Proof of 1|proof=<br />
By definition, <math>\ker f=\{g\in G\mid f(g)=e_H\}</math> where <math>e_H</math> is the identity of <math>H</math>. <math>\ker f</math> is a subgroup of <math>G</math> because:<br />
* '''Identity''': Since <math>f</math> is a homomorphism, <math>f(e_G)=e_H</math>. Therefore <math>e_G\in \ker f</math>, implying <math>\ker f</math> is non-empty and has an identity.<br />
* '''Closure''': Let <math>a,b\in \ker f</math>, then<br />
<math display="block">f(ab)=f(a)f(b)=e_He_H=e_H.</math><br />
Thus <math>ab\in \ker f</math>.<br />
* '''Inverses''': Let <math>a\in\ker f</math>, then <math display="block">f(a^{-1})=f(a)^{-1}=e_H^{-1}=e_H.</math><br />
Thus <math>a^{-1}\in \ker f</math>.<br />
<br />
Therefore, <math>\ker f</math> is a subgroup of <math>G</math>.<br />
<br />
Let <math>g\in G</math> and <math>k\in \ker f</math>, then<br />
<math display="block">f\left(gkg^{-1}\right)=f(g)f(k)f\left(g^{-1}\right)=f(g)e_Hf\left(g^{-1}\right)=f(g)f\left(g^{-1}\right)=f(g)f(g)^{-1}=e_H,</math><br />
thus <math>gkg^{-1}\in \ker f</math>. <br />
<br />
Therefore, <math>\ker f\trianglelefteq G</math> is a normal subgroup.<br />
}}<br />
<br />
{{Proof|title=Proof of 2|proof=<br />
By definition, <math>\operatorname{im} f=\{h\in H\mid \exists g\in G: f(g)=h\}</math>. <math>\operatorname{im} f</math> is a subgroup of <math>H</math> because:<br />
* '''Identity''': Since <math>f</math> is a homomorphism, <math>f(e_G)=e_H</math>. Therefore <math>e_H\in \operatorname{im} f</math>, implying <math>\operatorname{im} f</math> is non-empty and has an identity.<br />
* '''Closure''': Let <math>h_1,h_2\in \operatorname{im} f</math>, then by definition, there exists <math>g_1,g_2\in G</math> such that <math>f(g_1)=h_1</math> and <math>f(g_2)=h_2</math>. Thus<br />
<math display="block">h_1h_2=f(g_1)f(g_2)=f(g_1g_2).</math><br />
Thus <math>h_1h_2\in \operatorname{im} f</math>.<br />
* '''Inverses''': Let <math>h\in\operatorname{im}f</math>, and <math>g\in G</math> such that <math>f(g)=h</math>. Then <math display="block">h^{-1}=\left(f(g)\right)^{-1}=f\left(g^{-1}\right).</math><br />
Thus <math>h^{-1}\in \operatorname{im} f</math>.<br />
<br />
Therefore, <math>\operatorname{im} f< H</math> is a subgroup.<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=First_isomorphism_theorem&diff=95First isomorphism theorem2026-04-10T12:56:14Z<p>InfernalAtom683: </p>
<hr />
<div>The '''first isomorphism theorem''' is a fundamental result in [[abstract algebra]] that describes the relationship between a [[homomorphism]], its [[Kernel (algebra)|kernel]], and its [[Image (mathematics)|image]]. The theorem appears uniformly across algebraic structures such as [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], and [[Module (mathematics)|modules]], and serves as a prototype for many structural results in algebra. Specifically, given a homeomorphism, the quotient of its domain by its kernel is isomorphic to its image.<br />
<br />
== Group theory ==<br />
<br />
=== Statement ===<br />
Let <math>G</math> and <math>H</math> be [[Group|groups]] and <math>f\colon g\to H</math> a group homomorphism. Then,<br />
<br />
* The kernel of <math>f</math>, <math>\ker f\trianglelefteq G</math> is a normal subgroup of <math>G</math>.<br />
* The image of <math>f</math>, <math>\operatorname{im}f<G</math> is a subgroup of <math>H</math>.<br />
<br />
* <math>G/\ker f \cong \operatorname{im} f</math>.</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Equivalence_relation&diff=94Equivalence relation2026-04-10T12:53:57Z<p>InfernalAtom683: </p>
<hr />
<div>An '''equivalence relation''' is a [[binary relation]] on a [[set]] that groups elements into categories<ref group="Note">Not to be confused with [[category]] in [[category theory]].</ref> in which all members are considered "equivalent" under some criterion.<br />
<br />
== Definition ==<br />
A [[relation]] <math>\sim</math> on set <math>X</math> is a equivalence relation if it satisfies the following properties:<br />
<br />
* '''Reflexivity''': <math>\forall x\in X</math>, <math>x\sim x</math>.<br />
* '''Symmetry''': <math>\forall x,y\in X</math> such that <math>x\sim y</math>, <math>y\sim x</math>.<br />
* '''Transitivity''': <math>\forall x,y,z</math>, if <math>x\sim y</math> and <math>y\sim z</math>, then <math>x\sim z</math>.<br />
<br />
When <math>x\sim y</math>, "<math>x</math> is said to be equivalent to <math>y</math>" under the relation <math>\sim</math>.<br />
<br />
== Equivalence classes ==<br />
Given <math>x \in X</math>, the '''equivalence class''' of <math>x</math>, denoted <math display="block">[x]= \{y \in X \mid y \sim x\}</math> is the set of elements that are equivalent to <math>x</math>.<br />
== Notes ==<br />
<references group="Note" /><br />
<br />
== See also ==<br />
* [[Relation]]<br />
* [[Quotient set]]<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | equivalence relation | relation d'équivalence | Äquivalenzrelation | 等价关系 | 等價關係 | 同値関係 }}<br />
{{Terminology_table/row | equivalence class | classe d'équivalence | Äquivalenzklasse | 等价类 | 等價類 | 同値類 }}<br />
{{Terminology_table/row | equivalence | équivalence | Äquivalenz | 等价 | 等價 | 同値 }}<br />
{{Terminology_table/row | reflexive | réflexif | reflexiv | 自反的 | 自反的 | 反射的 }}<br />
{{Terminology_table/row | symmetric | symétrique | symmetrisch | 对称的 | 對稱的 | 対称的 }}<br />
{{Terminology_table/row | transitive | transitif | transitiv | 传递的 | 傳遞的 | 推移的 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Equivalence_relation&diff=93Equivalence relation2026-04-10T12:52:48Z<p>InfernalAtom683: </p>
<hr />
<div>An '''equivalence relation''' is a [[binary relation]] on a [[set]] that groups elements into categories<ref>Not to be confused with [[category]] in [[category theory]].</ref> in which all members are considered "equivalent" under some criterion.<br />
<br />
== Definition ==<br />
A [[relation]] <math>\sim</math> on set <math>X</math> is a equivalence relation if it satisfies the following properties:<br />
<br />
* '''Reflexivity''': <math>\forall x\in X</math>, <math>x\sim x</math>.<br />
* '''Symmetry''': <math>\forall x,y\in X</math> such that <math>x\sim y</math>, <math>y\sim x</math>.<br />
* '''Transitivity''': <math>\forall x,y,z</math>, if <math>x\sim y</math> and <math>y\sim z</math>, then <math>x\sim z</math>.<br />
<br />
When <math>x\sim y</math>, "<math>x</math> is said to be equivalent to <math>y</math>" under the relation <math>\sim</math>.<br />
<br />
== Equivalence classes ==<br />
Given <math>x \in X</math>, the '''equivalence class''' of <math>x</math>, denoted <math display="block">[x]= \{y \in X \mid y \sim x\}</math> is the set of elements that are equivalent to <math>x</math>.<br />
== Notes ==<br />
<references group="Note" /><br />
<br />
== See also ==<br />
* [[Relation]]<br />
* [[Quotient set]]<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | equivalence relation | relation d'équivalence | Äquivalenzrelation | 等价关系 | 等價關係 | 同値関係 }}<br />
{{Terminology_table/row | equivalence class | classe d'équivalence | Äquivalenzklasse | 等价类 | 等價類 | 同値類 }}<br />
{{Terminology_table/row | equivalence | équivalence | Äquivalenz | 等价 | 等價 | 同値 }}<br />
{{Terminology_table/row | reflexive | réflexif | reflexiv | 自反的 | 自反的 | 反射的 }}<br />
{{Terminology_table/row | symmetric | symétrique | symmetrisch | 对称的 | 對稱的 | 対称的 }}<br />
{{Terminology_table/row | transitive | transitif | transitiv | 传递的 | 傳遞的 | 推移的 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Darboux_integral&diff=92Darboux integral2026-04-10T11:09:38Z<p>InfernalAtom683: </p>
<hr />
<div>The '''Darboux integral''' is a formulation of integration in [[real analysis]] defined using upper and lower sums over partitions of an interval. It provides an order-theoretic approach to integration and is equivalent to the [[Riemann integral]].<br />
<br />
== Definition ==<br />
<br />
=== Darboux sums ===<br />
Let <math>f:[a,b]\to\mathbb{R}</math> be a bounded function. Let <math display="block"><br />
P=\{x_0,x_1,\dots,x_n\}, \quad a=x_0 < x_1 < \cdots < x_n=b<br />
</math> be a partition of the interval <math>[a,b]</math>.<br />
<br />
For each subinterval <math>[x_{i-1},x_i]</math> of <math>P</math>, define:<br />
<br />
*the infimum:<br />
<br />
<math display="block"><br />
m_i = \inf_{x\in[x_{i-1},x_i]}f(x),<br />
</math><br />
<br />
*the supremum:<br />
<br />
<math display="block"><br />
M_i = \sup_{x\in[x_{i-1},x_i]}f(x).<br />
</math><br />
<br />
The '''lower Darboux sum''' of <math>f</math> with respect to <math>P</math> is<br />
<math display="block"><br />
L(f,P)=\sum_{i=1}^n m_i(x_{i}-x_{i-1}),<br />
</math><br />
<br />
and the '''upper Darboux sum''' is<br />
<math display="block"><br />
U(f,P)=\sum_{i=1}^n M_i(x_{i}-x_{i-1}).<br />
</math><br />
<br />
=== The Darboux integral ===<br />
Let <math>\mathcal{P}([a,b])</math> denote the set of all partitions of <math>[a,b]</math>.<br />
<br />
The '''lower Darboux integral''' of <math>f</math> on <math>[a,b]</math> is defined by <math display="block">\underline{\int_a^b}f=\sup_{P\in\mathcal{P}([a,b])}L(f,P).</math><br />
<br />
and the '''upper Darboux integral''' is defined by <math display="block">\overline{\int_a^b}f=\inf_{P\in\mathcal{P}([a,b])}U(f,P).</math><br />
<br />
If the upper and lower Darboux integrals are equal, then the '''Darboux integral''' of <math>f</math> on <math>[a,b]</math> is defined by their common value, that is,<br />
<math display="block">\int_a^b f=\underline{\int_a^b}f=\overline{\int_a^b}f.</math><br />
In this case, function <math>f</math> is said to be '''Darboux-integrable'''.<br />
<br />
== See also ==<br />
<br />
* [[Riemann integral]]<br />
* [[Partition]]<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | Darboux integral | intégrale de Darboux | Darboux-Integral | Darboux 积分 | Darboux 積分 | ダルブー積分 }}<br />
{{Terminology_table/row | partition | partition | Partition | 分割 | 分割 | 分割 }}<br />
{{Terminology_table/row | Darboux sum | somme de Darboux | Darboux-Summe | Darboux 和 | Darboux 和 | ダルブー和 }}<br />
{{Terminology_table/row | upper Darboux sum | somme supérieure de Darboux | obere Darboux-Summe | 上 Darboux 和 | 上 Darboux 和 | 上ダルブー和 }}<br />
{{Terminology_table/row | lower Darboux sum | somme inférieure de Darboux | untere Darboux-Summe | 下 Darboux 和 | 下 Darboux 和 | 下ダルブー和 }}<br />
{{Terminology_table/row | upper integral | intégrale supérieure | oberes Integral | 上 Darboux 积分 | 上 Darboux 積分 | 上ダルブー積分 }}<br />
{{Terminology_table/row | lower integral | intégrale inférieure | unteres Integral | 下 Darboux 积分 | 下 Darboux 積分 | 下ダルブー積分 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Darboux_integral&diff=91Darboux integral2026-04-10T11:08:22Z<p>InfernalAtom683: </p>
<hr />
<div>The '''Darboux integral''' is a formulation of integration in [[real analysis]] defined using upper and lower sums over partitions of an interval. It provides an order-theoretic approach to integration and is equivalent to the [[Riemann integral]].<br />
<br />
== Definition ==<br />
<br />
=== Darboux sums ===<br />
Let <math>f:[a,b]\to\mathbb{R}</math> be a bounded function. Let <math display="block"><br />
P=\{x_0,x_1,\dots,x_n\}, \quad a=x_0 < x_1 < \cdots < x_n=b<br />
</math> be a partition of the interval <math>[a,b]</math>.<br />
<br />
For each subinterval <math>[x_{i-1},x_i]</math> of <math>P</math>, define:<br />
<br />
*the infimum:<br />
<br />
<math display="block"><br />
m_i = \inf_{x\in[x_{i-1},x_i]}f(x),<br />
</math><br />
<br />
*the supremum:<br />
<br />
<math display="block"><br />
M_i = \sup_{x\in[x_{i-1},x_i]}f(x).<br />
</math><br />
<br />
The '''lower Darboux sum''' of <math>f</math> with respect to <math>P</math> is<br />
<math display="block"><br />
L(f,P)=\sum_{i=1}^n m_i(x_{i}-x_{i-1}),<br />
</math><br />
<br />
and the '''upper Darboux sum''' is<br />
<math display="block"><br />
U(f,P)=\sum_{i=1}^n M_i(x_{i}-x_{i-1}).<br />
</math><br />
<br />
=== The Darboux integral ===<br />
Let <math>\mathcal{P}([a,b])</math> denote the set of all partitions of <math>[a,b]</math>.<br />
<br />
The '''lower Darboux integral''' of <math>f</math> on <math>[a,b]</math> is defined by <math display="block">\underline{\int_a^b}f=\sup_{P\in\mathcal{P}([a,b])}L(f,P).</math><br />
<br />
and the '''upper Darboux integral''' is defined by <math display="block">\overline{\int_a^b}f=\inf_{P\in\mathcal{P}([a,b])}U(f,P).</math><br />
<br />
If the upper and lower Darboux integrals are equal, then the '''Darboux integral''' of <math>f</math> on <math>[a,b]</math> is defined by their common value, that is,<br />
<math display="block">\int_a^b f=\underline{\int_a^b}f=\overline{\int_a^b}f.</math><br />
In this case, function <math>f</math> is said to be '''Darboux-integrable'''.<br />
<br />
== See also ==<br />
<br />
* [[Riemann integral]]<br />
* [[Partition]]<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | Darboux integral | intégrale de Darboux | Darboux-Integral | Darboux 积分 | Darboux 積分 | ダルブー積分 }}<br />
{{Terminology_table/row | Darboux sum | somme de Darboux | Darboux-Summe | Darboux 和 | Darboux 和 | ダルブー和 }}<br />
{{Terminology_table/row | upper Darboux sum | somme supérieure de Darboux | obere Darboux-Summe | 上 Darboux 和 | 上 Darboux 和 | 上ダルブー和 }}<br />
{{Terminology_table/row | lower Darboux sum | somme inférieure de Darboux | untere Darboux-Summe | 下 Darboux 和 | 下 Darboux 和 | 下ダルブー和 }}<br />
{{Terminology_table/row | upper integral | intégrale supérieure | oberes Integral | 上 Darboux 积分 | 上 Darboux 積分 | 上ダルブー積分 }}<br />
{{Terminology_table/row | lower integral | intégrale inférieure | unteres Integral | 下 Darboux 积分 | 下 Darboux 積分 | 下ダルブー積分 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Darboux_integral&diff=90Darboux integral2026-04-10T11:07:57Z<p>InfernalAtom683: </p>
<hr />
<div>The '''Darboux integral''' is a formulation of integration in [[real analysis]] defined using upper and lower sums over partitions of an interval. It provides an order-theoretic approach to integration and is equivalent to the [[Riemann integral]].<br />
<br />
== Definition ==<br />
<br />
=== Darboux sums ===<br />
Let <math>f:[a,b]\to\mathbb{R}</math> be a bounded function. Let <math display="block"><br />
P=\{x_0,x_1,\dots,x_n\}, \quad a=x_0 < x_1 < \cdots < x_n=b<br />
</math> be a partition of the interval <math>[a,b]</math>.<br />
<br />
For each subinterval <math>[x_{i-1},x_i]</math> of <math>P</math>, define:<br />
<br />
* the infimum:<br />
<br />
<math display="block"><br />
m_i = \inf_{x\in[x_{i-1},x_i]}f(x),<br />
</math><br />
<br />
* the supremum:<br />
<br />
<math display="block"><br />
M_i = \sup_{x\in[x_{i-1},x_i]}f(x).<br />
</math><br />
<br />
The '''lower Darboux sum''' of <math>f</math> with respect to <math>P</math> is<br />
<math display="block"><br />
L(f,P)=\sum_{i=1}^n m_i(x_{i}-x_{i-1}),<br />
</math><br />
<br />
and the '''upper Darboux sum''' is<br />
<math display="block"><br />
U(f,P)=\sum_{i=1}^n M_i(x_{i}-x_{i-1}).<br />
</math><br />
<br />
=== The Darboux integral ===<br />
Let <math>\mathcal{P}([a,b])</math> denote the set of all partitions of <math>[a,b]</math>.<br />
<br />
The '''lower Darboux integral''' of <math>f</math> on <math>[a,b]</math> is defined by <math display="block">\underline{\int_a^b}f=\sup_{P\in\mathcal{P}([a,b])}L(f,P).</math><br />
<br />
and the '''upper Darboux integral''' is defined by <math display="block">\overline{\int_a^b}f=\inf_{P\in\mathcal{P}([a,b])}U(f,P).</math><br />
<br />
If the upper and lower Darboux integrals are equal, then the '''Darboux integral''' of <math>f</math> on <math>[a,b]</math> is defined by their common value, that is,<br />
<math display="block">\int_a^b f=\underline{\int_a^b}f=\overline{\int_a^b}f.</math><br />
In this case, function <math>f</math> is said to be '''Darboux-integrable'''.<br />
<br />
== See also ==<br />
<br />
* [[Riemann integral]]<br />
* [[Partition]]<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | Darboux integral | intégrale de Darboux | Darboux-Integral | Darboux 积分 | Darboux 積分 | ダルブー積分 }}<br />
{{Terminology_table/row | Darboux sum | somme de Darboux | Darboux-Summe | Darboux 和 | Darboux 和 | ダルブー和 }}<br />
{{Terminology_table/row | upper Darboux sum | somme supérieure de Darboux | obere Darboux-Summe | 上 Darboux 和 | 上 Darboux 和 | 上ダルブー和 }}<br />
{{Terminology_table/row | lower Darboux sum | somme inférieure de Darboux | untere Darboux-Summe | 下 Darboux 和 | 下 Darboux 和 | 下ダルブー和 }}<br />
{{Terminology_table/row | upper integral | intégrale supérieure | oberes Integral | 上 Darboux 积分 | 上 Darboux 積分 | 上ダルブー積分 }}<br />
{{Terminology_table/row | lower integral | intégrale inférieure | unteres Integral | 下 Darboux 积分 | 下 Darboux 積分 | 下ダルブー積分 }}<br />
}}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Locally_Euclidean_space&diff=89Locally Euclidean space2026-04-10T04:56:48Z<p>InfernalAtom683: </p>
<hr />
<div>A '''locally Euclidean space''' is a [[topological space]] that resembles a [[Euclidean space]] locally. Specifically, every point in a locally Euclidean space has an open neighborhood that is homeomorphic to an open subset in <math>\mathbb{R}^n</math>. The concept of locally Euclidean space is a central object in the definition of [[topological manifold]].<br />
<br />
== Definition ==<br />
A topological space <math>X</math> is locally Euclidean of dimension <math>n</math>, if for every point <math>x\in X</math>, there exists an open neighborhood <math>U</math> of <math>x</math>, and a homeomorphism <math display="block">\phi: U\to V</math>where <math>V\subset \mathbb{R}^n</math> is open with the standard topology on <math>\mathbb{R}^n</math>.<br />
<br />
The pair <math>(U,\phi)</math> is called a [[chart]] (or coordinate chart) on <math>X</math>. A collection of such charts that covers <math>X</math> is an [[atlas]].<br />
<br />
== See also ==<br />
<br />
* [[Euclidean space]]<br />
* [[Topological manifold]]<br />
* [[Homeomorphism]]<br />
* [[Chart]]<br />
* [[Atlas]]<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | locally Euclidean space | espace localement euclidien | lokal euklidischer Raum | 局部 Euclid 空间 | 局部 Euclid 空間 | 局所ユークリッド空間 }}<br />
{{Terminology_table/row | Euclidean space | espace euclidien | euklidischer Raum | Euclid 空间 | Euclid 空間 | ユークリッド空間 }}<br />
{{Terminology_table/row | chart | carte | Karte | 坐标卡 | 座標卡 | チャート }}<br />
{{Terminology_table/row | atlas | atlas | Atlas | 图册 | 圖冊 | アトラス }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Darboux_integral&diff=88Darboux integral2026-04-09T07:50:02Z<p>InfernalAtom683: </p>
<hr />
<div>The '''Darboux integral''' is a formulation of integration in [[real analysis]] defined using upper and lower sums over partitions of an interval. It provides an order-theoretic approach to integration and is equivalent to the [[Riemann integral]].<br />
<br />
== Definition ==<br />
<br />
=== Darboux sums ===<br />
Let <math>f:[a,b]\to\mathbb{R}</math> be a bounded function. Let <math display="block"><br />
P=\{x_0,x_1,\dots,x_n\}, \quad a=x_0 < x_1 < \cdots < x_n=b<br />
</math> be a partition of the interval <math>[a,b]</math>.<br />
<br />
For each subinterval <math>[x_{i-1},x_i]</math> of <math>P</math>, define:<br />
<br />
* the infimum:<br />
<br />
<math display="block"><br />
m_i = \inf_{x\in[x_{i-1},x_i]}f(x),<br />
</math><br />
<br />
* the supremum:<br />
<br />
<math display="block"><br />
M_i = \sup_{x\in[x_{i-1},x_i]}f(x).<br />
</math><br />
<br />
The '''lower Darboux sum''' of <math>f</math> with respect to <math>P</math> is<br />
<math display="block"><br />
L(f,P)=\sum_{i=1}^n m_i(x_{i}-x_{i-1}),<br />
</math><br />
<br />
and the '''upper Darboux sum''' is<br />
<math display="block"><br />
U(f,P)=\sum_{i=1}^n M_i(x_{i}-x_{i-1}).<br />
</math><br />
<br />
=== The Darboux integral ===<br />
Let <math>\mathcal{P}([a,b])</math> denote the set of all partitions of <math>[a,b]</math>.<br />
<br />
The '''lower Darboux integral''' of <math>f</math> on <math>[a,b]</math> is defined by <math display="block">\underline{\int_a^b}f=\sup_{P\in\mathcal{P}([a,b])}L(f,P).</math><br />
<br />
and the '''upper Darboux integral''' is defined by <math display="block">\overline{\int_a^b}f=\inf_{P\in\mathcal{P}([a,b])}U(f,P).</math><br />
<br />
If the upper and lower Darboux integrals are equal, then the '''Darboux integral''' of <math>f</math> on <math>[a,b]</math> is defined by their common value, that is,<br />
<math display="block">\int_a^b f=\underline{\int_a^b}f=\overline{\int_a^b}f.</math><br />
In this case, function <math>f</math> is said to be '''Darboux-integrable'''.<br />
<br />
== See also ==<br />
<br />
* [[Riemann integral]]<br />
* [[Partition]]</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homotopy&diff=87Homotopy2026-04-09T07:48:37Z<p>InfernalAtom683: </p>
<hr />
<div>A '''homotopy''' is a continuous deformation between two [[Continuous function|continuous functions]] from one [[topological space]] to another. Specifically, a homotopy between two functions is a continuous map that, for each point in the domain, provides a path from its image under the first function to its image under the second. If such a function exists between two functions, they are said to be homotopic.<br />
<br />
Intuitively, a homotopy is the continuous transformation of paths that varies over time. It shows how one function can be smoothly bent, stretched, or deformed into the other without tearing or folding.<br />
<br />
== Definition ==<br />
If <math>f,g:X\to Y</math> are continuous functions between topological spaces <math>X</math> and <math>Y</math>, a homotopy <math>H</math> from <math>f</math> to <math>g</math> is a continuous map<br />
<br />
<math display="block">H:[0,1] \times X \to Y</math><br />
<br />
such that for all <math>x \in X</math>:<br />
<br />
* <math>H_0(x) = f(x)</math>,<br />
<br />
* <math>H_1(x) = g(x)</math>.<br />
<br />
Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math><ref group="Note">Not to be confused with "<math>\cong</math>" for homeomorphism.</ref>.<br />
<br />
== Homotopy equivalence ==<br />
Two topological spaces <math>X</math> and <math>Y</math> are '''homotopy equivalent''' if there exist continuous maps<br />
<br />
<math display="block">f:X \to Y \quad \text{and} \quad g:Y \to X</math><br />
<br />
such that<br />
<br />
* <math>g \circ f \simeq \operatorname{id}_X</math>,<br />
<br />
* <math>f \circ g \simeq \operatorname{id}_Y</math>.<br />
<br />
In this case, <math>X</math> and <math>Y</math> have the same "essential shape" from the perspective of homotopy. This concept is distinct from [[homeomorphism]], which is a stricter condition requiring the maps to be inverses of each other. For instance, a solid [[sphere]] is homotopy equivalent to a single point, but they are not homeomorphic.<br />
<br />
== Note ==<br />
<references group="Note" /><br />
<br />
== See also ==<br />
<br />
* [[Homeomorphism]]<br />
* [[Continuous function]]<br />
* [[Homeotopy]]<br />
* [[Homotopy type theory]]<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homotopy | homotopie | Homotopie | 同伦 | 同倫 | ホモトピー }}<br />
{{Terminology_table/row | homotopic | homotope | homotop | 同伦的 | 同倫的 | ホモトピック }}<br />
{{Terminology_table/row | homotopy equivalence | équivalence d'homotopie | Homotopieäquivalenz | 同伦等价 | 同倫等價 | ホモトピー同値 }}<br />
{{Terminology_table/row | homotopy equivalent | homotopiquement équivalent | homotopieäquivalent | 同伦等价的 | 同倫等價的 | ホモトピー同値な }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homotopy&diff=86Homotopy2026-04-09T07:30:19Z<p>InfernalAtom683: </p>
<hr />
<div>A '''homotopy''' is a continuous deformation between two [[Continuous function|continuous functions]] from one [[topological space]] to another. Specifically, a homotopy between two functions is a continuous map that, for each point in the domain, provides a path from its image under the first function to its image under the second. If such a function exists between two functions, they are said to be homotopic.<br />
<br />
Intuitively, a homotopy is the continuous transformation of paths that varies over time. It shows how one function can be smoothly bent, stretched, or deformed into the other without tearing or folding.<br />
<br />
== Definition ==<br />
If <math>f,g:X\to Y</math> are continuous functions between topological spaces <math>X</math> and <math>Y</math>, a homotopy <math>H</math> from <math>f</math> to <math>g</math> is a continuous map<br />
<br />
<math display="block">H:[0,1] \times X \to Y</math><br />
<br />
such that for all <math>x \in X</math>:<br />
<br />
* <math>H_0(x) = f(x)</math>,<br />
<br />
* <math>H_1(x) = g(x)</math>.<br />
<br />
Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math>.<br />
<br />
== Homotopy equivalence ==<br />
Two topological spaces <math>X</math> and <math>Y</math> are '''homotopy equivalent''' if there exist continuous maps<br />
<br />
<math display="block">f:X \to Y \quad \text{and} \quad g:Y \to X</math><br />
<br />
such that<br />
<br />
* <math>g \circ f \simeq \operatorname{id}_X</math>,<br />
<br />
* <math>f \circ g \simeq \operatorname{id}_Y</math>.<br />
<br />
In this case, <math>X</math> and <math>Y</math> have the same "essential shape" from the perspective of homotopy. This concept is distinct from [[homeomorphism]], which is a stricter condition requiring the maps to be inverses of each other. For instance, a solid [[sphere]] is homotopy equivalent to a single point, but they are not homeomorphic.<br />
<br />
== See also ==<br />
<br />
* [[Homeomorphism]]<br />
* [[Continuous function]]<br />
* [[Homeotopy]]<br />
* [[Homotopy type theory]]<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homotopy | homotopie | Homotopie | 同伦 | 同倫 | ホモトピー }}<br />
{{Terminology_table/row | homotopic | homotope | homotop | 同伦的 | 同倫的 | ホモトピック }}<br />
{{Terminology_table/row | homotopy equivalence | équivalence d'homotopie | Homotopieäquivalenz | 同伦等价 | 同倫等價 | ホモトピー同値 }}<br />
{{Terminology_table/row | homotopy equivalent | homotopiquement équivalent | homotopieäquivalent | 同伦等价的 | 同倫等價的 | ホモトピー同値な }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Quotient_group&diff=85Quotient group2026-04-09T07:28:12Z<p>InfernalAtom683: </p>
<hr />
<div>A '''quotient group''' is a [[group]] obtained by aggregating similar elements of a larger group using an [[equivalence relation]] that preserves some of the group structure.<br />
<br />
== Definitions ==<br />
Let <math>G</math> be a group and <math>N\trianglelefteq G</math> a [[normal subgroup]].<br />
<br />
=== Definition via cosets ===<br />
The quotient group <math>G/N</math> is the set of left [[Coset|cosets]] <math display="block">G/N:=\{gN\mid g\in G\}</math>with group operation <math>(gN)(hN):=(gh)N.</math><br />
<br />
=== Definition via equivalence relations ===<br />
Define a [[relation]] on <math>G</math> by<math display="block">g\sim h\Longleftrightarrow g^{-1}h\in N.</math><br />
<br />
The relation <math>\sim</math> is a equivalence relation, and denote <math>[g]:=\{a\mid a\sim g\}</math> as the equivalence class of <math>g</math>. The quotient group is defined by the set of [[Equivalence relation#Equivalence classes|equivalence classes]]<math display="block">G/N:=G/\sim</math>with operation <math>[g][h]:=[gh].</math></div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Mathepedia:About&diff=84Mathepedia:About2026-04-09T07:26:37Z<p>InfernalAtom683: </p>
<hr />
<div>== About Mathepedia ==<br />
Mathepedia is a professional mathematical encyclopedia dedicated to pure mathematics. Aimed to create a public space for mathematical concept explanation and discussion, Mathepedia is designed to serve as a structured and reliable reference for mathematical concepts, definitions, and theories.<br />
<br />
== Mission ==<br />
Mathepedia seeks to provide a resource that is logically precise, conceptually clear, and mathematically rigorous. We believe that all mathematical ideas — from the foundational arithmetic operations to the well-combined calculus — deserve to be documented with precision and quality. Rather than merely listing results, Mathepedia aims to clarify relationships between concepts, highlight underlying ideas, and support a deeper understanding of mathematical theory.<br />
<br />
== Scope ==<br />
Mathepedia focuses exclusively on pure mathematics, including but not limited to:<br />
* Algebra: group theory, ring theory, field theory, linear algebra<br />
* Analysis: real and complex analysis, functional analysis<br />
* Geometry and Topology: differential geometry, algebraic topology, general topology<br />
* Number Theory: elementary and analytic number theory<br />
* Foundations: logic, set theory, category theory<br />
<br />
The emphasis is on theory, structure, and abstraction, rather than applications or historical narrative.<br />
<br />
== Rigor and rendering ==<br />
All content is written with a commitment to mathematical rigor and consistency. Definitions are given in precise form where possible, often accompanied by equivalent formulations or categorical interpretations when appropriate.<br />
<br />
The render of formulas on different client devices '''should not be a problem.''' Therefore, Mathematical expressions are typeset using standard [https://www.latex-project.org/ LaTeX], ensuring clarity and uniformity across platforms through [https://www.mathjax.org/ MathJax] rendering.<br />
<br />
== Licensing ==<br />
Mathepedia is released under the CC-BY-SA (Creative Commons Attribution-ShareAlike) licence. This permits redistribution and adaptation of the material, provided proper attribution is given and derivative works are shared under the same terms.</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Hausdorff_space&diff=78Hausdorff space2026-04-06T03:43:50Z<p>InfernalAtom683: </p>
<hr />
<div>A '''Hausdorff space''' (or '''<math>T_2</math> space''') is a type of [[topological space]] in which points can be "cleanly separated" by neighborhoods. Specifically, for any two distinct points, there exist disjoint [[Open set|open sets]] containing each point. Consequently, Hausdorff property ensures that limits of sequences are unique when they exist.<br />
<br />
== Definitions ==<br />
A topological space <math>(X,\tau)</math> is Hausdorff, if for any two points <math>x,y\in X</math>, there exists two disjoint open sets <math>U,V\in \tau</math>, <math>U\cap V=\varnothing</math>, such that <math>x\in U</math> and <math>y\in V</math>.<br />
<br />
=== Equivalent Definitions ===<br />
Any convergent [[sequence]] in <math>X</math> has at most one limit.<br />
<br />
== Properties ==<br />
{{Property|property=Subspaces of Hausdorff spaces are Hausdorff.}}{{Proof|proof=Let <math>Y \subseteq X</math> with <math>X</math> Hausdorff. For <math>y_1, y_2 \in Y, y_1 \neq y_2</math>, there exist disjoint open sets <math>U, V \subseteq X</math> with <math>y_1 \in U</math> and <math>y_2 \in V</math>. Then <math>U \cap Y</math> and <math>V \cap Y</math> are disjoint open sets in <math>Y</math> containing <math>y_1</math> and <math>y_2</math>.}}{{Property|property=Finite products of Hausdorff spaces are Hausdorff.}}{{Proof|proof=Let <math>X_1, \dots, X_n</math> be Hausdorff. Consider points <math>(x_1, \dots, x_n) \neq (y_1, \dots, y_n)</math>. There exists an index <math>i</math> with <math>x_i \neq y_i</math>.<br />
<br />
Since <math>X_i</math> is Hausdorff, choose disjoint open sets <math>U_i, V_i \subseteq X_i</math> containing <math>x_i</math> and <math>y_i</math>. Then <math>U_1 \times \dots \times U_n</math> and <math>V_1 \times \dots \times V_n</math> are disjoint open sets containing the two points.}}{{Property|property=Compact subsets of Hausdorff spaces are closed.}}{{Proof|proof=Let <math>K \subseteq X</math> be compact and <math>X</math> Hausdorff. For any <math>x \in X \setminus K</math>, for each <math>y \in K</math> choose disjoint open sets <math>U_y\ni x</math> and <math>V_y \ni y</math>. <br />
<br />
The collection <math>\{V_y | y \in K\}</math> covers <math>K</math>. By compactness, finitely many <math>V_{y_1}, \dots, V_{y_n}</math> cover <math>K</math>. Then <math display="block">U = \bigcap_{i=1}^n U_{y_i}</math> is open, contains <math>x</math>, and is disjoint from <math>K</math>. Hence <math>K</math> is closed.}}<br />
<br />
== Examples ==<br />
Every metric space is a Hausdorff space.<br />
{{Proof|proof=<div style="float:right; width:320px; margin:0 0 0.5em 1em;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=8pt]{standalone}<br />
\usetikzlibrary{calc}<br />
\usepackage{amsmath}<br />
\begin{document}<br />
\begin{tikzpicture}[scale=1.3]<br />
<br />
\def\r{1.5}<br />
<br />
\coordinate (x) at (0,0);<br />
<br />
\def\ang{20}<br />
<br />
\coordinate (y) at ({2*\r*cos(\ang)}, {2*\r*sin(\ang)});<br />
<br />
\draw[fill=blue!25!cyan, opacity=0.2, dashed, thick] (x) circle (\r);<br />
\draw[fill=blue!25!cyan, opacity=0.2, dashed, thick] (y) circle (\r);<br />
<br />
\fill (x) circle (2pt);<br />
\fill (y) circle (2pt);<br />
<br />
\node[below left] at (x) {$x$};<br />
\node[above right] at (y) {$y$};<br />
<br />
\draw[black, dashed, thick]<br />
(x) -- ({\r*cos(\ang)}, {\r*sin(\ang)})<br />
node[midway, above, sloped] {\footnotesize $r=\dfrac{d(x,y)}{2}$};<br />
<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki></div><br />
<br />
Let <math>(X,d)</math> be a metric space, take two distinct points <math>x,y</math> such that <math>d(x,y)>0</math>.<br />
<br />
Consider open balls <math>B(x,r)</math> and <math>B(y,r)</math>where <math>r=\frac{d(x,y)}{2}>0</math>. The open balls are both open with <math>x\in B(x,r)</math> and <math>y\in B(y,r)</math>.<br />
<br />
For contradiction, assume there exists <math>z\in B(x,r)\cap B(y,r)</math>, then <math>d(x,z)<r</math> and <math>d(y,z)<r</math>. By triangular inequality, <math display="block">d(x,y)\leq d(x,y)+d(x,z)<r+r=d(x,y),</math> there exists a contradiction.<br />
<br />
Thus, <math>B(x,r)\cap B(y,r)=\varnothing</math>; therefore <math>(X,d)</math> is Hausdorff.}}<br />
<br />
== See also ==<br />
<br />
* [[Topological space]]<br />
* [[Convergence]]<br />
<br />
==Terminology==<br />
{{Terminology_table|<br />
{{Terminology_table/row | Hausdorff space | espace de Hausdorff (espace séparé) | hausdorff-Raum (hausdorffscher Raum) | Hausdorff 空间 | Hausdorff 空間 | ハウスドルフ空間 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Locally_Euclidean_space&diff=77Locally Euclidean space2026-04-06T03:37:15Z<p>InfernalAtom683: </p>
<hr />
<div>A '''locally Euclidean space''' is a [[topological space]] that resembles a [[Euclidean space]] locally. Specifically, every point in a locally Euclidean space has an open neighborhood that is homeomorphic to an open subset in <math>\mathbb{R}^n</math>. The concept of locally Euclidean space is a central object in the definition of [[topological manifold]].<br />
<br />
== Definition ==<br />
A topological space <math>X</math> is locally Euclidean of dimension <math>n</math>, if for every point <math>x\in X</math>, there exists an open neighborhood <math>U</math> of <math>x</math>, and a homeomorphism <math display="block">\phi: U\to V</math>where <math>V\subset \mathbb{R}^n</math> is open with the standard topology on <math>\mathbb{R}^n</math>.<br />
<br />
== See also ==<br />
<br />
* [[Euclidean space]]<br />
* [[Topological manifold]]<br />
* [[Homeomorphism]]<br />
* [[Chart]]<br />
* [[Atlas]]<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | locally Euclidean space | espace localement euclidien | lokal euklidischer Raum | 局部 Euclid 空间 | 局部 Euclid 空間 | 局所ユークリッド空間 }}<br />
{{Terminology_table/row | Euclidean space | espace euclidien | euklidischer Raum | Euclid 空间 | Euclid 空間 | ユークリッド空間 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homotopy&diff=76Homotopy2026-04-06T03:28:38Z<p>InfernalAtom683: </p>
<hr />
<div>A '''homotopy''' is a continuous deformation between two [[Continuous function|continuous functions]] from one [[topological space]] to another. Specifically, a homotopy between two functions is a continuous map that, for each point in the domain, provides a path from its image under the first function to its image under the second. If such a function exists between two functions, they are said to be homotopic.<br />
<br />
Intuitively, a homotopy is the continuous transformation of paths that varies over time. It shows how one function can be smoothly bent, stretched, or deformed into the other without tearing or folding.<br />
<br />
== Definition ==<br />
If <math>f,g:X\to Y</math> are continuous functions between topological spaces <math>X</math> and <math>Y</math>, a homotopy <math>H</math> from <math>f</math> to <math>g</math> is a continuous map<br />
<br />
<math display="block">H:X \times [0,1] \to Y</math><br />
<br />
such that for all <math>x \in X</math>:<br />
<br />
* <math>H(x,0) = f(x)</math>,<br />
<br />
* <math>H(x,1) = g(x)</math>.<br />
<br />
Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math>.<br />
<br />
== Homotopy equivalence ==<br />
Two topological spaces <math>X</math> and <math>Y</math> are '''homotopy equivalent''' if there exist continuous maps<br />
<br />
<math display="block">f:X \to Y \quad \text{and} \quad g:Y \to X</math><br />
<br />
such that<br />
<br />
* <math>g \circ f \simeq \operatorname{id}_X</math>,<br />
<br />
* <math>f \circ g \simeq \operatorname{id}_Y</math>.<br />
<br />
In this case, <math>X</math> and <math>Y</math> have the same "essential shape" from the perspective of homotopy. This concept is distinct from [[homeomorphism]], which is a stricter condition requiring the maps to be inverses of each other. For instance, a solid [[sphere]] is homotopy equivalent to a single point, but they are not homeomorphic.<br />
<br />
== See also ==<br />
<br />
* [[Homeomorphism]]<br />
* [[Continuous function]]<br />
* [[Homeotopy]]<br />
* [[Homotopy type theory]]<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homotopy | homotopie | Homotopie | 同伦 | 同倫 | ホモトピー }}<br />
{{Terminology_table/row | homotopic | homotope | homotop | 同伦的 | 同倫的 | ホモトピック }}<br />
{{Terminology_table/row | homotopy equivalence | équivalence d'homotopie | Homotopieäquivalenz | 同伦等价 | 同倫等價 | ホモトピー同値 }}<br />
{{Terminology_table/row | homotopy equivalent | homotopiquement équivalent | homotopieäquivalent | 同伦等价的 | 同倫等價的 | ホモトピー同値な }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Category_of_topological_spaces&diff=75Category of topological spaces2026-04-06T03:26:24Z<p>InfernalAtom683: </p>
<hr />
<div>The '''category of topological spaces''', denoted <math>\mathsf{Top}</math> or <math>\mathbf{Top}</math>, is the [[category]] whose objects are [[Topological space|topological spaces]] and whose [[Morphism|morphisms]] are [[Continuous function|continuous functions]].<br />
<br />
== Definition ==<br />
The category <math>\mathsf{Top}</math> consists of:<br />
<br />
* <math>\operatorname{ob}(\mathsf{Top})</math> is the class of all topological spaces;<br />
* <math>\operatorname{mor}(\mathsf{Top})</math> is the class of all continuous functions between topological spaces;<br />
* The composition operation <math>\circ</math> is given by the composition of ordinary functions.<br />
<br />
== Subcategories ==<br />
Several important subcategories of <math>\mathsf{Top}</math> arise by restricting objects:<br />
<br />
* <math>\mathsf{Haus}</math>, [[category of Hausdorff spaces]];<br />
* <math>\mathsf{Comp}</math>, [[category of Compact spaces]];<br />
<br />
* <math>\mathsf{Top}_{*}</math>, [[category of pointed topological spaces]];<br />
* <math>\mathsf{CW}</math>, [[category of CW complexes]] or cell complexes.<br />
<br />
== See also ==<br />
<br />
* [[Category theory]]<br />
* [[Morphism]]<br />
* [[Category of sets]]<br />
* [[Category of groups]]</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Quotient_group&diff=74Quotient group2026-04-05T08:18:51Z<p>InfernalAtom683: Created page with "A '''quotient group''' is a group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure. == Definitions == Let <math>G</math> be a group and <math>N\trianglelefteq G</math> a normal subgroup. === Definition via cosets === The quotient group <math>G/N</math> is the set of left cosets <math display="block">G/N:=\{gN\mid g\in G\}</math>with group operation <math>(gN)(hN):=(gh)N.</math> === Defi..."</p>
<hr />
<div>A '''quotient group''' is a group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure.<br />
<br />
== Definitions ==<br />
Let <math>G</math> be a group and <math>N\trianglelefteq G</math> a normal subgroup.<br />
<br />
=== Definition via cosets ===<br />
The quotient group <math>G/N</math> is the set of left cosets <math display="block">G/N:=\{gN\mid g\in G\}</math>with group operation <math>(gN)(hN):=(gh)N.</math><br />
<br />
=== Definition via equivalence relations ===<br />
Define a relation on <math>G</math> by<math display="block">g\sim h\Longleftrightarrow g^{-1}h\in N.</math><br />
<br />
The relation <math>\sim</math> is a equivalence relation, and denote <math>[g]:=\{a\mid a\sim g\}</math> as the equivalence class of <math>g</math>. The quotient group is defined by the set of equivalence classes<math display="block">G/N:=G/\sim</math>with operation <math>[g][h]:=[gh].</math></div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homeomorphism&diff=73Homeomorphism2026-04-04T11:43:39Z<p>InfernalAtom683: </p>
<hr />
<div>[[File:Topology joke.jpg|thumb|250x250px|A homeomorphism that turns a coffee mug into a donut continuously.]]<br />
A '''homeomorphism''' is a special type of [[function]] between two [[Topological space|topological spaces]], that establishes that the two spaces are fundamentally the same from a topological perspective. Specifically, it is a [[Continuous function|continuous]] [[bijective]] function whose [[inverse function]] is also continuous. Homeomorphisms are the [[Isomorphism|isomorphisms]] in the [[category of topological spaces]] <math>\mathsf{Top}</math>, which preserves all [[topological properties]] of a topological space. If such a function exists between two spaces, they are said to be '''homeomorphic'''. <br />
<br />
Intuitively, two spaces are homeomorphic if one can be continuously deformed into the other by stretching, bending, and twisting, without cutting, tearing, or gluing. A typical intuitive example is that a mug with a handle is homeomorphic to a donut. This concept is distinct from [[Homotopy#Homotopy equivalence|homotopy equivalence]], which allows deformations that involve collapsing. For instance, a solid ball can be continuously shrunk to a point by a homotopy, but such a deformation is not a homeomorphism because it is not bijective and the inverse would not be continuous.<br />
<br />
== Definitions ==<br />
A function <math>f</math> between topological spaces <math>X</math> and <math>Y</math> is called a '''homeomorphism''', if:<br />
<br />
* <math>f</math> is continuous,<br />
* <math>f</math> is bijective,<br />
* <math>f^{-1}</math> is continuous.<br />
<br />
Two topological spaces <math>X</math> and <math>Y</math> are called '''homeomorphic''' if there exists a homeomorphism between them, denoted <math>X\cong Y</math>.<br />
<br />
=== Equivalent Definitions ===<br />
A homeomorphism is a bijection that is continuous and [[Open function|open]], or continuous and [[Closed function|closed]].<br />
<br />
== Properties ==<br />
{{Property|property=The composition of two homeomorphisms is again a homeomorphism.}}{{Proof|proof=Let <math>f: X \to Y</math> and <math>g: Y \to Z</math> be homeomorphisms. Then:<br />
<br />
* <math>g \circ f: X \to Z</math> is bijective, since the composition of two bijections is a bijection.<br />
<br />
* <math>g \circ f</math> is continuous, as the composition of two continuous functions.<br />
<br />
* The inverse is <math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}</math>, which is continuous because it is the composition of the continuous functions <math>g^{-1}</math> and <math>f^{-1}</math>.<br />
<br />
Thus <math>g \circ f</math> satisfies all requirements of a homeomorphism.}}<br />
<br />
<br />
{{Property|property=The inverse of a homeomorphism is again a homeomorphism.}}<br />
{{Proof|proof=Let <math>f: X \to Y</math> be a homeomorphism. Then:<br />
* <math>f^{-1}</math> is continuous by definition,<br />
* <math>f^{-1}</math> is bijective, since the inverse of a bijection is again a bijection,<br />
* <math>\left(f^{-1}\right)^{-1}=f</math> is continuous by definition.}}<br />
<br />
<br />
{{Property|property=Homeomorphism is an [[equivalence relation]].}}<br />
<br />
{{Proof|proof=<br />
* '''Reflexivity''': The identity map <math>\operatorname{id}_X:X\to X</math> is a continuous bijection on any topological space <math>X</math>, whose inverse is itself. Thus <math>\operatorname{id}_X</math> is a homeomorphism.<br />
* '''Symmetry''': If <math>f: X\to Y</math> is a homeomorphism, then its inverse <math>f^{-1}: Y\to X</math> is again a homeomorphism.<br />
* '''Transitivity''': If <math>f: X\to Y</math> and <math>g: Y\to Z</math> are homeomorphisms, then <math>g\circ f: X\to Z</math> is again a homeomorphism.<br />
}}<br />
<br />
== Examples ==<br />
<br />
=== Open interval ===<br />
The [[open interval]] <math>(0,1)</math> is homeomorphic to <math>\mathbb{R}</math>.<br />
{{Proof|proof=The map <math>f:(0,1)\to \mathbb{R}</math> defined by<br />
<math display="block">f(x)=\tan\left(\pi\left(x-\dfrac12\right)\right)</math><br />
is a homeomorphism. Indeed, <math>f</math> is continuous because it is a composition of continuous functions. The restriction <math>\tan:(-\pi/2,\pi/2)\to\mathbb{R}</math> is bijective with continuous inverse <math>\arctan:\mathbb{R}\to(-\pi/2,\pi/2)</math>. Therefore <math>f</math> is bijective and its inverse<br />
<math display="block">f^{-1}(y)=\dfrac1\pi\arctan(y)+\dfrac12</math><br />
is continuous. Thus <math>f</math> is a homeomorphism.}}<br />
<br />
=== Stereographic projection ===<br />
The [[Euclidean plane]] <math>\mathbb{R}^2</math> is homeomorphic to the [[2-sphere]] minus one point, denoted <math>S^2 \setminus \{N\}</math> where <math>N=(0,0,1)</math> is the [[north pole]].<br />
<br />
{{Proof|proof=<br />
<div style="float:right; width:320px; margin:0 0 0.5em 1em;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=15pt]{standalone}<br />
\usepackage{tikz-3dplot}<br />
\usetikzlibrary{calc, arrows.meta}<br />
\begin{document}<br />
\def\viewTheta{70}<br />
\def\viewPhi{20}<br />
\tdplotsetmaincoords{\viewTheta}{\viewPhi}<br />
<br />
\begin{tikzpicture}[tdplot_main_coords, scale=2, line cap=round, line join=round]<br />
<br />
\def\R{1}<br />
\coordinate (O) at (0,0,0);<br />
\coordinate (N) at (0,0,\R);<br />
<br />
\def\thetaS{60}<br />
\def\phiS{30}<br />
\pgfmathsetmacro{\px}{\R * sin(\thetaS) * cos(\phiS)}<br />
\pgfmathsetmacro{\py}{\R * sin(\thetaS) * sin(\phiS)}<br />
\pgfmathsetmacro{\pz}{\R * cos(\thetaS)}<br />
\coordinate (P) at (\px, \py, \pz);<br />
<br />
\pgfmathsetmacro{\ux}{\px / (1 - \pz)}<br />
\pgfmathsetmacro{\uy}{\py / (1 - \pz)}<br />
\coordinate (Pprime) at (\ux, \uy, 0);<br />
<br />
\pgfmathsetmacro{\eqFrontStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\eqBackStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqBackEnd}{\viewPhi+180}<br />
<br />
\pgfmathsetmacro{\cotViewTheta}{cos(\viewTheta)/sin(\viewTheta)}<br />
\pgfmathsetmacro{\cotThetaS}{cos(\thetaS)/sin(\thetaS)}<br />
\pgfmathsetmacro{\cosAlpha}{max(min(-\cotThetaS * \cotViewTheta, 1), -1)}<br />
\pgfmathsetmacro{\alpha}{acos(\cosAlpha)}<br />
<br />
\pgfmathsetmacro{\latFrontStart}{\viewPhi-180}<br />
\pgfmathsetmacro{\latFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\latBackStart}{\viewPhi}<br />
\pgfmathsetmacro{\latBackEnd}{\viewPhi+180}<br />
<br />
\draw[thick, black] (-1.2,0,0) -- (0,0,0);<br />
\draw[thick, black] (0,-3,0) -- (0,0,0);<br />
\draw[thick, black] (0,0,-1.2) -- (0,0,0);<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (2.2,0,0) node[anchor=north east]{$x$};<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (0,3.0,0) node[anchor=north west]{$y$};<br />
\draw[thick, dashed] (0,0,0) -- (N);<br />
\draw[thick, ->, >=Stealth] (N) -- (0,0,1.8) node[anchor=south]{$z$};<br />
\begin{scope}[tdplot_screen_coords]<br />
\shade[ball color=cyan, opacity=0.15] (0,0) circle (\R);<br />
\draw[cyan!60!blue, thick] (0,0) circle (\R);<br />
\end{scope}<br />
<br />
\tdplotdrawarc[cyan!60!blue, thick]{(O)}{\R}{\eqFrontStart}{\eqFrontEnd}{}{}<br />
\tdplotdrawarc[cyan!60!blue, thin, dashed]{(O)}{\R}{\eqBackStart}{\eqBackEnd}{}{}<br />
<br />
\pgfmathsetmacro{\rLat}{\R * sin(\thetaS)}<br />
\coordinate (CenterLat) at (0,0,\pz);<br />
<br />
\tdplotsetrotatedcoords{0}{0}{0}<br />
\tdplotsetrotatedcoordsorigin{(CenterLat)}<br />
\tdplotdrawarc[cyan!70!black, thin, dashed, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latBackStart}{\latBackEnd}{}{}<br />
\tdplotdrawarc[cyan!70!black, thin, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latFrontStart}{\latFrontEnd}{}{}<br />
\tdplotsetrotatedcoordsorigin{(O)}<br />
<br />
\draw[red, thick, dashed] (N) -- (P);<br />
\draw[red, thick, ->, >=Stealth] (P) -- (Pprime);<br />
<br />
\fill[black] (N) circle (0.8pt) node[anchor=south east] {$N$};<br />
\fill[red] (P) circle (1pt) node[anchor=south west, text=black] {$(x,y,z)$};<br />
\fill[red] (Pprime) circle (1pt) node[anchor=north west, text=black] {$p(x,y,z) = (u,v)$};<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
Define the [[stereographic projection]] <math>p: S^2 \setminus \{N\} \to \mathbb{R}^2</math> by<br />
<math display="block">p(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right).</math><br />
This map is continuous because it is a rational function with denominator nonzero (since <math>z<1</math> on <math>S^2\setminus\{N\}</math>).<br />
<br />
The inverse map <math>p^{-1}: \mathbb{R}^2 \to S^2 \setminus \{N\}</math> is given by<br />
<math display="block">p^{-1}(u,v) = \left( \frac{2u}{u^2+v^2+1}, \frac{2v}{u^2+v^2+1}, \frac{u^2+v^2-1}{u^2+v^2+1} \right).</math><br />
This is also continuous as a composition of continuous functions. One verifies that <math>p \circ p^{-1} = \text{id}_{\mathbb{R}^2}</math> and <math>p^{-1} \circ p = \operatorname{id}_{S^2\setminus\{N\}}</math> by direct substitution. Hence <math>p</math> is a homeomorphism.<br />
}}<br />
<br />
=== Quotient space ===<br />
The unit interval <math>[0,1]</math> with the endpoints identified (the quotient space <math>[0,1]/\sim</math> where <math>0\sim 1</math>) is homeomorphic to the circle <math>S^1</math>.<br />
<br />
{{Proof|proof=Define the map <math>f:[0,1] \to S^1</math> by <math display="block">f(t)=(\cos(2\pi t), \sin(2\pi t)).</math> This map is continuous and [[Surjection|surjective]], and satisfies <math>f(0)=f(1)=(1,0)</math>.<br />
<br />
Consider the equivalence relation <math>\sim</math>, and let <math>q:[0,1]\to [0,1]/\sim</math> be the [[quotient map]]. By the [[universal property]] of the quotient map, there exists a unique continuous map <math>\tilde{f}: [0,1]/\sim \to S^1</math> such that <math>\tilde{f} \circ q = f</math>; that is, the following diagram commutes.<br />
<div style="text-align: center;"><br />
<br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
{[0,1]} && {S^1} \\<br />
& {[0,1]/{\sim}} \arrow["f", from=1-1, to=1-3]<br />
\arrow["q"', from=1-1, to=2-2]<br />
\arrow["{\exists! \tilde{f}}"', dashed, from=2-2, to=1-3]<br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
The map <math>\tilde{f}</math> is bijective because:<br />
* Surjectivity follows from surjectivity of <math>f</math>;<br />
* [[Injection|Injectivity]] holds because <math display="block">\tilde{f}([t])=\tilde{f}([s])\Rightarrow t=s \text{ or } \{t,s\}=\{0,1\},</math> but in the latter case <math>[t]=[s]</math> in the quotient. <br />
<br />
Hence <math>\tilde{f}</math> is a continuous bijection.<br />
<br />
The space <math>[0,1]/\sim</math> is compact as the quotient of a [[compact space]], and <math>S^1</math> is [[Hausdorff space|Hausdorff]]. By the [[Compact-to-Hausdorff theorem]], a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.<br />
<br />
Therefore <math>\tilde{f}</math> is a homeomorphism.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz">\documentclass[tikz,border=10pt]{standalone}<br />
\usepackage{amsmath}<br />
\usetikzlibrary{arrows.meta,calc}<br />
\begin{document}<br />
<br />
\begin{tikzpicture}[<br />
>={Stealth[scale=1.1]},<br />
dot/.style={circle,fill=black,inner sep=1.6pt},<br />
label text/.style={font=\Large,align=center}<br />
]<br />
<br />
\def\r{1.4}<br />
\def\gap{50}<br />
<br />
\coordinate (C1) at (0,0);<br />
\coordinate (C2) at (5.5,0);<br />
\coordinate (C3) at (11,0);<br />
<br />
\begin{scope}[shift={(C1)}]<br />
\draw[thick] (-\r,0) coordinate (A) -- (\r,0) coordinate (B);<br />
\node[dot] at (A) {};<br />
\node[dot] at (B) {};<br />
\node[label text] at (0,-2.6) {$[0,1]$};<br />
\coordinate (R1) at (\r,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C1)+(2,0)$) -- ($(C2)+(-2,0)$)<br />
node[midway,above] {$q$};<br />
<br />
\begin{scope}[shift={(C2)}]<br />
\draw[thick]<br />
(180-\gap:\r)<br />
arc[start angle=180-\gap,end angle=360+\gap,radius=\r];<br />
<br />
\node[dot] (L) at (180-\gap:\r) {};<br />
\node[dot] (R) at (\gap:\r) {};<br />
<br />
\draw[dashed,->,thick]<br />
(L) .. controls +(0,0) and +(-0.8,-0.1) .. (90:\r);<br />
\draw[dashed,->,thick]<br />
(R) .. controls +(0,0) and +(0.8,-0.1) .. (90:\r);<br />
<br />
\node[label text] at (0,-2.6)<br />
{$[0,1]/\sim$ \\[-0.4ex]\normalsize $(0\sim1)$};<br />
<br />
\coordinate (R2) at (2,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C2)+(2,0)$) -- ($(C3)+(-2,0)$)<br />
node[midway,above] {$\overset{\tilde{f}}{\cong}$};<br />
<br />
\begin{scope}[shift={(C3)}]<br />
\draw[thick] (0,0) circle (\r);<br />
\node[dot] at (90:\r) {};<br />
\node[label text] at (0,-2.6) {$S^{1}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}</kroki><br />
</div><br />
<br />
}}<br />
<br />
== Counterexamples ==<br />
<math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>.<br />
<br />
{{Proof|proof=For contradiction, suppose that there exists a homeomorphism <math>f: \mathbb{R}\to\mathbb{R}^2</math>.<br />
<br />
Consider the subspace <math>\mathbb{R}\setminus\{0\}</math> of <math>\mathbb{R}</math>. The [[restriction]] on it, <math>\left.f\right|_{\mathbb{R}\setminus\{0\}}: \mathbb{R}\setminus\{0\}\to \mathbb{R}^2\setminus\{f(0)\}</math> is also a homeomorphism.<br />
<br />
However, <math>\mathbb{R}\setminus\{0\}</math> has two connected components, <math>(-\infty,0)</math> and <math>(0,\infty)</math>, while <math>\mathbb{R}^2\setminus\{f(0)\}</math> is connected, which contradicts the assumption that the two spaces are homeomorphic.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz, border=10pt]{standalone}<br />
\usepackage{amsmath, amssymb}<br />
<br />
\begin{document}<br />
\begin{tikzpicture}<br />
\begin{scope}[xshift=-5cm]<br />
\draw[thick] (-3, 0) -- (3, 0);<br />
<br />
\filldraw[fill=white, draw=black, thick] (0, 0) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2.5, 0) -- (-1, 0);<br />
\fill[cyan!60!blue] (-2.5, 0) circle (2pt);<br />
\fill[cyan!60!blue] (-1, 0) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (1.3, 0) -- (2.2, 0);<br />
\fill[cyan!60!blue] (1.3, 0) circle (2pt);<br />
\fill[cyan!60!blue] (2.2, 0) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -2) {$\mathbb{R} \setminus \{0\}$};<br />
\end{scope}<br />
<br />
\begin{scope}[xshift=4cm]<br />
\draw[thick] (-3.5, -2.5) -- (3.5, -2.5) -- (3.5, 2.5) -- (-3.5, 2.5) -- cycle;<br />
<br />
\filldraw[fill=white, draw=black, thick] (-0.3, -0.2) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2, 1.5) .. controls (-0.5, 1) and (-0.8, -1) .. (-1.2, -1.8);<br />
\fill[cyan!60!blue] (-2, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (-1.2, -1.8) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 1.5) .. controls (1.5, 1.8) and (2.5, 1) .. (2, 0.5);<br />
\fill[cyan!60!blue] (-0.3, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (2, 0.5) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 0.5) .. controls (1, 0) and (1, -1) .. (2.5, -2);<br />
\fill[cyan!60!blue] (-0.3, 0.5) circle (2pt);<br />
\fill[cyan!60!blue] (2.5, -2) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.5, -0.8) .. controls (0, -1.8) and (1, -1.5) .. (0.8, -1);<br />
\fill[cyan!60!blue] (-0.5, -0.8) circle (2pt);<br />
\fill[cyan!60!blue] (0.8, -1) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -3.5) {$\mathbb{R}^2 \setminus \{f(0)\}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
Hence, no such homeomorphism exists; therefore <math>\mathbb{R}</math> is not homeomorphic to <math>\mathbb{R}^2</math>}}The map from the interval <math>[0,1)</math> to the 1-sphere <math>S^1</math>,<br />
<math display="block">\phi: [0,1)\to S^1,\quad x\mapsto e^{2\pi ix}</math><br />
is continuous and bijective, but not a homeomorphism. <br />
{{Proof|proof=<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass{article}<br />
\usepackage{tikz}<br />
\usetikzlibrary{arrows.meta}<br />
<br />
\begin{document}<br />
<br />
\begin{tikzpicture}<br />
\draw (0,0) -- (4,0);<br />
<br />
\draw (0.15, 0.25) -- (0, 0.25) -- (0, -0.25) -- (0.15, -0.25);<br />
<br />
\draw (3.9, 0.25) to[bend left=45] (3.9, -0.25);<br />
<br />
\draw[-{Stealth[length=3mm, width=2mm]}] (4.5, 1.2) to[bend left=30] node[above=2pt] {$\phi$} (7.0, 1.2);<br />
<br />
\draw (9.5, 0) circle (2);<br />
<br />
\begin{scope}[rotate around={90:(11.5,0)}]<br />
\draw (11.39, 0.25) to[bend left=45] (11.39, -0.25);<br />
\draw (11.5, -0.25) -- (11.5, 0.25);<br />
\draw (11.5, 0.25) -- (11.65, 0.25);<br />
\draw (11.5, -0.25) -- (11.65, -0.25);<br />
<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
The map <math>\phi</math> is:<br />
* Continuous, as it is the composition of continuous maps <math>x\mapsto 2\pi x</math> and <math>t\mapsto e^{it}</math>.<br />
* Injective, because if <math>e^{2\pi i x_1}=e^{2\pi i x_2}</math>, then <math>x_1-x_2\in \mathbb{Z}</math>. Since <math>x_1,x_2\in [0,1)</math>, it follows that <math>x_1=x_2</math>.<br />
* Surjective, since every point of <math>S^1</math> can be written as <math>e^{2\pi i x}</math> for some <math>x\in [0,1)</math>.<br />
Hence <math>\phi</math> is a continuous bijection.<br />
<br />
However, <math>\phi</math> is not a homeomorphism.<br />
<br />
Consider the sequence<math display="block">z_n = e^{2\pi i (1-\tfrac{1}{n})} \in S^1.</math><br />
Then <math>z_n \to 1 = e^{2\pi i \cdot 0}</math> in <math>S^1</math>. But<br />
<math display="block">\phi^{-1}(z_n) = 1-\tfrac{1}{n} \to 1,</math><br />
which does not converge to <math>\phi^{-1}(1)=0</math> in <math>[0,1)</math>. Thus <math>\phi^{-1}</math> is not continuous.<br />
}}<br />
== Topological invariants ==<br />
A [[topological invariant]] is a property of a topological space that is preserved under homeomorphisms. In other words, if two spaces are homeomorphic, they either both possess the property or both do not. Invariants are the important tools to classify topological spaces. If two spaces differ in any topological invariant, they cannot be homeomorphic. Conversely, showing that two spaces share many invariants is often the first step on proving they are homeomorpic, though it is never sufficient by itself.<br />
<br />
=== Common topological invariants ===<br />
<br />
* [[Connectedness]]<br />
* [[Compactness]]<br />
* [[Hausdorff space|Hausdorff property]]<br />
* [[Cardinality]] of the space<br />
<br />
=== Algebraic invariants ===<br />
More powerful invariants come from [[algebraic topology]], which assigns algebraic objects to topological spaces.<br />
<br />
* [[Fundamental group]]<br />
* [[Homology group]]<br />
* [[Higher homotopy group]]<br />
<br />
==Homeomorphism group==<br />
The collection of all [[autohomeomorphisms]] of a topological space <math>X</math> forms a [[group]] under composition operation, known as the homeomorphism group of <math>X</math>, denoted <math>\operatorname{Homeo}(X)</math>. The homeomorphism group captures the symmetry in topology. It describes the ways in which a topological space can be continuously transformed onto itself.<br />
<br />
The homeomorphism group <math>\operatorname{Homeo}(X)</math> is a faithful [[group action]] on its underlying set <math>X</math>. It moves points in <math>X</math> continuously onto <math>X</math> itself, and the topological structure of <math>X</math> is also reflected in the algebraic invariants such as the [[Orbit|orbits]] and [[Stabilizer|stabilizers]] of the action.<br />
<br />
For example, consider the 2-sphere <math>S^2</math> as a thin rubber membrane tightly wraped around a ball. Each autohomeomorphism of <math>S^2</math>, which is an element in <math>\operatorname{Homeo}(S^2)</math>, corresponds to a continuous deformation of this membrane. This operation can be stretching, bending, twisting, or any composition of these operations, so the rubber always remains attached to the ball.<br />
<div style="text-align: center;"><br />
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</div><br />
Under the natural action of <math>\operatorname{Homeo}(S^2)</math>, every point on the sphere can be moved continuously to any other point. This example shows how the homeomorphism group captures the symmetry of a topological space in the perspective of continuity.<br />
<br />
==See also==<br />
* [[Homotopy]]<br />
* [[Topology]]<br />
* [[Homeomorphism group]]<br />
<br />
[[Category:Topopogy]]<br />
<br />
==Terminology==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homeomorphism | homéomorphisme | Homeomorphismus | 同胚 | 同胚 | 同相写像 }}<br />
{{Terminology_table/row | homeomorphic | homéomorphe | homeomorph | 同胚的 | 同胚的 | 同相 }}<br />
{{Terminology_table/row | topological invariant | invariant topologique | topologische Invariante | 拓扑不变量 | 拓撲不變量 | 位相不変量 }}<br />
{{Terminology_table/row | autohomeomorphism | autohoméomorphisme | Selbsthomöomorphismus | 自同胚 | 自同胚 | 自己同相写像 }}<br />
{{Terminology_table/row | homeomorphism group | groupe des homéomorphismes | Homöomorphismengruppe | 同胚群 | 同胚群 | 同相群 }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homotopy&diff=72Homotopy2026-04-04T11:30:27Z<p>InfernalAtom683: </p>
<hr />
<div>A '''homotopy''' is a continuous deformation between two [[Continuous function|continuous functions]] from one [[topological space]] to another. Specifically, a homotopy between two functions is a continuous map that, for each point in the domain, provides a path from its image under the first function to its image under the second. If such a function exists between two functions, they are said to be homotopic.<br />
<br />
Intuitively, a homotopy is the continuous transformation of paths that varies over time. It shows how one function can be smoothly bent, stretched, or deformed into the other without tearing or folding.<br />
<br />
== Definition ==<br />
If <math>f,g:X\to Y</math> are continuous functions between topological spaces <math>X</math> and <math>Y</math>, a homotopy <math>H</math> from <math>f</math> to <math>g</math> is a continuous map<br />
<br />
<math display="block">H:X \times [0,1] \to Y</math><br />
<br />
such that for all <math>x \in X</math>:<br />
<br />
* <math>H(x,0) = f(x)</math>,<br />
<br />
* <math>H(x,1) = g(x)</math>.<br />
<br />
Two continuous functions <math>f</math> and <math>g</math> are called homotopic if there exists a homotopy between them, denoted <math>f\simeq g</math>.<br />
<br />
== Homotopy equivalence ==<br />
Two topological spaces <math>X</math> and <math>Y</math> are '''homotopy equivalent''' if there exist continuous maps<br />
<br />
<math display="block">f:X \to Y \quad \text{and} \quad g:Y \to X</math><br />
<br />
such that<br />
<br />
* <math>g \circ f \simeq \operatorname{id}_X</math>,<br />
<br />
* <math>f \circ g \simeq \operatorname{id}_Y</math>.<br />
<br />
In this case, <math>X</math> and <math>Y</math> have the same "essential shape" from the perspective of homotopy. This concept is distinct from [[homeomorphism]], which is a stricter condition requiring the maps to be inverses of each other. For instance, a solid [[sphere]] is homotopy equivalent to a single point, but they are not homeomorphic.<br />
<br />
== Terminology ==<br />
{{Terminology_table|<br />
{{Terminology_table/row | homotopy | homotopie | Homotopie | 同伦 | 同倫 | ホモトピー }}<br />
{{Terminology_table/row | homotopic | homotope | homotop | 同伦的 | 同倫的 | ホモトピック }}<br />
{{Terminology_table/row | homotopy equivalence | équivalence d'homotopie | Homotopieäquivalenz | 同伦等价 | 同倫等價 | ホモトピー同値 }}<br />
{{Terminology_table/row | homotopy equivalent | homotopiquement équivalent | homotopieäquivalent | 同伦等价的 | 同倫等價的 | ホモトピー同値な }}<br />
}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Template:Polyglot/row&diff=71Template:Polyglot/row2026-04-04T11:29:08Z<p>InfernalAtom683: InfernalAtom683 moved page Template:Polyglot/row to Template:Terminology table/row</p>
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<div>#REDIRECT [[Template:Terminology table/row]]</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Template:Terminology_table/row&diff=70Template:Terminology table/row2026-04-04T11:29:08Z<p>InfernalAtom683: InfernalAtom683 moved page Template:Polyglot/row to Template:Terminology table/row</p>
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| {{{6|{{{ja}}}}}}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Template:Polyglot&diff=69Template:Polyglot2026-04-04T11:29:08Z<p>InfernalAtom683: InfernalAtom683 moved page Template:Polyglot to Template:Terminology table</p>
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<div>#REDIRECT [[Template:Terminology table]]</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Template:Terminology_table&diff=68Template:Terminology table2026-04-04T11:29:08Z<p>InfernalAtom683: InfernalAtom683 moved page Template:Polyglot to Template:Terminology table</p>
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|}</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Template:Terminology_table/row&diff=67Template:Terminology table/row2026-04-04T11:27:39Z<p>InfernalAtom683: Created page with "|- | '''{{{1|{{{en}}}}}}''' | {{{2|{{{fr}}}}}} | {{{3|{{{de}}}}}} | {{{4|{{{zh-cn|{{{zh-hans|}}}}}}}}} | {{{5|{{{zh-tw|{{{zh-hant|}}}}}}}}} | {{{6|{{{ja}}}}}}"</p>
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<div>|-<br />
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<hr />
<div>The '''first isomorphism theorem''' is a fundamental result in [[abstract algebra]] that describes the relationship between a [[homomorphism]], its [[Kernel (algebra)|kernel]], and its [[Image (mathematics)|image]]. The theorem appears uniformly across algebraic structures such as [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], and [[Module (mathematics)|modules]], and serves as a prototype for many structural results in algebra. Specifically, given a homeomorphism, the quotient of its domain by its kernel is isomorphic to its image.<br />
<br />
== Group theory ==<br />
<br />
=== Statement ===<br />
Let <math>G</math> and <math>H</math> be [[Group|groups]] and <math>f\colon g\to H</math> a group homomorphism. Then,<br />
<br />
* The kernel of <math>f</math> is a normal subgroup of <math>G</math>.<br />
* The image of <math>f</math> is a subgroup of <math>H</math>.<br />
<br />
* <math>G/\ker f \cong \operatorname{im} f</math>.</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=First_isomorphism_theorem&diff=64First isomorphism theorem2026-04-04T11:18:24Z<p>InfernalAtom683: Created page with "The '''first isomorphism theorem''' is a fundamental result in abstract algebra that describes the relationship between a homomorphism, its kernel, and its image. The theorem appears uniformly across algebraic structures such as groups, rings, and modules, and serves as a prototype for many structural results in algebra. Specifically, given a homeo..."</p>
<hr />
<div>The '''first isomorphism theorem''' is a fundamental result in [[abstract algebra]] that describes the relationship between a [[homomorphism]], its [[Kernel (algebra)|kernel]], and its [[Image (mathematics)|image]]. The theorem appears uniformly across algebraic structures such as [[Group (mathematics)|groups]], [[Ring (mathematics)|rings]], and [[Module (mathematics)|modules]], and serves as a prototype for many structural results in algebra. Specifically, given a homeomorphism, the quotient of its domain by its kernel is isomorphic to its image.<br />
<br />
== Group theory ==<br />
<br />
=== Statement ===<br />
Let <math>G</math> and <math>H</math> be [[Group|groups]] and <math>f\colon g\to H</math> a group homomorphism. Then,<br />
<br />
* The kernel of <math>f</math> is a normal subgroup of <math>G</math>.<br />
* The image of <math>f</math> is a subgroup of <math>H</math>.<br />
<br />
* <math>G/\ker f \cong \operatorname{im} f</math> .</div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Template:Main_Page/Category_ring&diff=63Template:Main Page/Category ring2026-04-04T10:18:26Z<p>InfernalAtom683: </p>
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<div style="position: absolute; top: 17%; left: 17%; width: 66%; height: 66%; background-color: #ffffff; border-radius: 50%;"></div><br />
<div style="position: absolute; left: 50.0%; top: 8.5%; transform: translate(-50%, -50%); font-size: 16px; text-transform: capitalize; z-index: 10;">[[Algebra|<span style="color:#fff;">'''Algebra'''</span>]]</div><br />
<div style="position: absolute; left: 23.3%; top: 18.2%; transform: translate(-50%, -50%); font-size: 16px; text-transform: capitalize; z-index: 10;">[[Foundation|<span style="color:#fff;">'''Foundation'''</span>]]</div><br />
<div style="position: absolute; left: 9.1%; top: 42.8%; transform: translate(-50%, -50%); font-size: 16px; text-transform: capitalize; z-index: 10;">[[Discrete|<span style="color:#fff;">'''Discrete'''</span>]]</div><br />
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<div><div style="position: relative; width: 600px; height: 600px; margin: 0 auto;"><br />
<div style="position: absolute; top: 0; left: 0; width: 100%; height: 100%; border-radius: 50%; transform: rotate(20deg); background: conic-gradient( transparent 0deg 2deg, #c1a10d 2deg 38deg, transparent 38deg 42deg, #50b61a 42deg 78deg, transparent 78deg 82deg, #008740 82deg 118deg, transparent 118deg 122deg, ##2bb09d 122deg 158deg, transparent 158deg 162deg, #2f95b6 162deg 198deg, transparent 198deg 202deg, #3366bb 202deg 238deg, transparent 238deg 242deg, #6420c2 242deg 278deg, transparent 278deg 282deg, #c80fa2 282deg 318deg, transparent 318deg 322deg, #cc0000 322deg 358deg, transparent 358deg 360deg );"></div><br />
<div style="position: absolute; top: 17%; left: 17%; width: 66%; height: 66%; background-color: #ffffff; border-radius: 50%;"></div><br />
<div style="position: absolute; left: 50.0%; top: 8.5%; transform: translate(-50%, -50%); font-size: 16px; text-transform: capitalize; z-index: 10;">[[Algebra|<span style="color:#fff;">'''Algebra'''</span>]]</div><br />
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</div><br />
</div></div>InfernalAtom683https://www.mathepedia.wiki/w/index.php?title=Homeomorphism&diff=61Homeomorphism2026-04-01T07:42:34Z<p>InfernalAtom683: /* Examples */</p>
<hr />
<div>[[File:Topology joke.jpg|thumb|250x250px|A homeomorphism that turns a coffee mug into a donut continuously.]]<br />
A '''homeomorphism''' is a special type of [[function]] between two [[Topological space|topological spaces]], that establishes that the two spaces are fundamentally the same from a topological perspective. Specifically, it is a [[Continuous function|continuous]] [[bijective]] function whose [[inverse function]] is also continuous. Homeomorphisms are the [[Isomorphism|isomorphisms]] in the [[category of topological spaces]] <math>\mathsf{Top}</math>, which preserves all [[topological properties]] of a topological space. If such a function exists between two spaces, they are said to be '''homeomorphic'''. <br />
<br />
Intuitively, two spaces are homeomorphic if one can be continuously deformed into the other by stretching, bending, and twisting, without cutting, tearing, or gluing. A typical intuitive example is that a mug with a handle is homeomorphic to a donut. This concept is distinct from [[Homotopy#Homotopy equivalence|homotopy equivalence]], which allows deformations that involve collapsing. For instance, a solid ball can be continuously shrunk to a point by a homotopy, but such a deformation is not a homeomorphism because it is not bijective and the inverse would not be continuous.<br />
<br />
== Definitions ==<br />
A function <math>f</math> between topological spaces <math>X</math> and <math>Y</math> is called a '''homeomorphism''', if:<br />
<br />
* <math>f</math> is continuous,<br />
* <math>f</math> is bijective,<br />
* <math>f^{-1}</math> is continuous.<br />
<br />
Two topological spaces <math>X</math> and <math>Y</math> are called '''homeomorphic''' if there exists a homeomorphism between them, denoted <math>X\cong Y</math>.<br />
<br />
=== Equivalent Definitions ===<br />
A homeomorphism is a bijection that is continuous and [[Open function|open]], or continuous and [[Closed function|closed]].<br />
<br />
== Properties ==<br />
{{Property|property=The composition of two homeomorphisms is again a homeomorphism.}}{{Proof|proof=Let <math>f: X \to Y</math> and <math>g: Y \to Z</math> be homeomorphisms. Then:<br />
<br />
* <math>g \circ f: X \to Z</math> is bijective, since the composition of two bijections is a bijection.<br />
<br />
* <math>g \circ f</math> is continuous, as the composition of two continuous functions.<br />
<br />
* The inverse is <math>(g \circ f)^{-1} = f^{-1} \circ g^{-1}</math>, which is continuous because it is the composition of the continuous functions <math>g^{-1}</math> and <math>f^{-1}</math>.<br />
<br />
Thus <math>g \circ f</math> satisfies all requirements of a homeomorphism.}}<br />
<br />
<br />
{{Property|property=The inverse of a homeomorphism is again a homeomorphism.}}<br />
{{Proof|proof=Let <math>f: X \to Y</math> be a homeomorphism. Then:<br />
* <math>f^{-1}</math> is continuous by definition,<br />
* <math>f^{-1}</math> is bijective, since the inverse of a bijection is again a bijection,<br />
* <math>\left(f^{-1}\right)^{-1}=f</math> is continuous by definition.}}<br />
<br />
<br />
{{Property|property=Homeomorphism is an [[equivalence relation]].}}<br />
<br />
{{Proof|proof=<br />
* '''Reflexivity''': The identity map <math>\operatorname{id}_X:X\to X</math> is a continuous bijection on any topological space <math>X</math>, whose inverse is itself. Thus <math>\operatorname{id}_X</math> is a homeomorphism.<br />
* '''Symmetry''': If <math>f: X\to Y</math> is a homeomorphism, then its inverse <math>f^{-1}: Y\to X</math> is again a homeomorphism.<br />
* '''Transitivity''': If <math>f: X\to Y</math> and <math>g: Y\to Z</math> are homeomorphisms, then <math>g\circ f: X\to Z</math> is again a homeomorphism.<br />
}}<br />
<br />
== Examples ==<br />
<br />
=== Open interval ===<br />
The [[open interval]] <math>(0,1)</math> is homeomorphic to <math>\mathbb{R}</math>.<br />
{{Proof|proof=The map <math>f:(0,1)\to \mathbb{R}</math> defined by<br />
<math display="block">f(x)=\tan\left(\pi\left(x-\dfrac12\right)\right)</math><br />
is a homeomorphism. Indeed, <math>f</math> is continuous because it is a composition of continuous functions. The restriction <math>\tan:(-\pi/2,\pi/2)\to\mathbb{R}</math> is bijective with continuous inverse <math>\arctan:\mathbb{R}\to(-\pi/2,\pi/2)</math>. Therefore <math>f</math> is bijective and its inverse<br />
<math display="block">f^{-1}(y)=\dfrac1\pi\arctan(y)+\dfrac12</math><br />
is continuous. Thus <math>f</math> is a homeomorphism.}}<br />
<br />
=== Stereographic projection ===<br />
The [[Euclidean plane]] <math>\mathbb{R}^2</math> is homeomorphic to the [[2-sphere]] minus one point, denoted <math>S^2 \setminus \{N\}</math> where <math>N=(0,0,1)</math> is the [[north pole]].<br />
<br />
{{Proof|proof=<br />
<div style="float:right; width:320px; margin:0 0 0.5em 1em;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz,border=15pt]{standalone}<br />
\usepackage{tikz-3dplot}<br />
\usetikzlibrary{calc, arrows.meta}<br />
\begin{document}<br />
\def\viewTheta{70}<br />
\def\viewPhi{20}<br />
\tdplotsetmaincoords{\viewTheta}{\viewPhi}<br />
<br />
\begin{tikzpicture}[tdplot_main_coords, scale=2, line cap=round, line join=round]<br />
<br />
\def\R{1}<br />
\coordinate (O) at (0,0,0);<br />
\coordinate (N) at (0,0,\R);<br />
<br />
\def\thetaS{60}<br />
\def\phiS{30}<br />
\pgfmathsetmacro{\px}{\R * sin(\thetaS) * cos(\phiS)}<br />
\pgfmathsetmacro{\py}{\R * sin(\thetaS) * sin(\phiS)}<br />
\pgfmathsetmacro{\pz}{\R * cos(\thetaS)}<br />
\coordinate (P) at (\px, \py, \pz);<br />
<br />
\pgfmathsetmacro{\ux}{\px / (1 - \pz)}<br />
\pgfmathsetmacro{\uy}{\py / (1 - \pz)}<br />
\coordinate (Pprime) at (\ux, \uy, 0);<br />
<br />
\pgfmathsetmacro{\eqFrontStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\eqBackStart}{\viewPhi - 180}<br />
\pgfmathsetmacro{\eqBackEnd}{\viewPhi+180}<br />
<br />
\pgfmathsetmacro{\cotViewTheta}{cos(\viewTheta)/sin(\viewTheta)}<br />
\pgfmathsetmacro{\cotThetaS}{cos(\thetaS)/sin(\thetaS)}<br />
\pgfmathsetmacro{\cosAlpha}{max(min(-\cotThetaS * \cotViewTheta, 1), -1)}<br />
\pgfmathsetmacro{\alpha}{acos(\cosAlpha)}<br />
<br />
\pgfmathsetmacro{\latFrontStart}{\viewPhi-180}<br />
\pgfmathsetmacro{\latFrontEnd}{\viewPhi}<br />
\pgfmathsetmacro{\latBackStart}{\viewPhi}<br />
\pgfmathsetmacro{\latBackEnd}{\viewPhi+180}<br />
<br />
\draw[thick, black] (-1.2,0,0) -- (0,0,0);<br />
\draw[thick, black] (0,-3,0) -- (0,0,0);<br />
\draw[thick, black] (0,0,-1.2) -- (0,0,0);<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (2.2,0,0) node[anchor=north east]{$x$};<br />
\draw[thick, ->, >=Stealth] (0,0,0) -- (0,3.0,0) node[anchor=north west]{$y$};<br />
\draw[thick, dashed] (0,0,0) -- (N);<br />
\draw[thick, ->, >=Stealth] (N) -- (0,0,1.8) node[anchor=south]{$z$};<br />
\begin{scope}[tdplot_screen_coords]<br />
\shade[ball color=cyan, opacity=0.15] (0,0) circle (\R);<br />
\draw[cyan!60!blue, thick] (0,0) circle (\R);<br />
\end{scope}<br />
<br />
\tdplotdrawarc[cyan!60!blue, thick]{(O)}{\R}{\eqFrontStart}{\eqFrontEnd}{}{}<br />
\tdplotdrawarc[cyan!60!blue, thin, dashed]{(O)}{\R}{\eqBackStart}{\eqBackEnd}{}{}<br />
<br />
\pgfmathsetmacro{\rLat}{\R * sin(\thetaS)}<br />
\coordinate (CenterLat) at (0,0,\pz);<br />
<br />
\tdplotsetrotatedcoords{0}{0}{0}<br />
\tdplotsetrotatedcoordsorigin{(CenterLat)}<br />
\tdplotdrawarc[cyan!70!black, thin, dashed, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latBackStart}{\latBackEnd}{}{}<br />
\tdplotdrawarc[cyan!70!black, thin, tdplot_rotated_coords]{(0,0,0)}{\rLat}{\latFrontStart}{\latFrontEnd}{}{}<br />
\tdplotsetrotatedcoordsorigin{(O)}<br />
<br />
\draw[red, thick, dashed] (N) -- (P);<br />
\draw[red, thick, ->, >=Stealth] (P) -- (Pprime);<br />
<br />
\fill[black] (N) circle (0.8pt) node[anchor=south east] {$N$};<br />
\fill[red] (P) circle (1pt) node[anchor=south west, text=black] {$(x,y,z)$};<br />
\fill[red] (Pprime) circle (1pt) node[anchor=north west, text=black] {$p(x,y,z) = (u,v)$};<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
Define the [[stereographic projection]] <math>p: S^2 \setminus \{N\} \to \mathbb{R}^2</math> by<br />
<math display="block">p(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right).</math><br />
This map is continuous because it is a rational function with denominator nonzero (since <math>z<1</math> on <math>S^2\setminus\{N\}</math>).<br />
<br />
The inverse map <math>p^{-1}: \mathbb{R}^2 \to S^2 \setminus \{N\}</math> is given by<br />
<math display="block">p^{-1}(u,v) = \left( \frac{2u}{u^2+v^2+1}, \frac{2v}{u^2+v^2+1}, \frac{u^2+v^2-1}{u^2+v^2+1} \right).</math><br />
This is also continuous as a composition of continuous functions. One verifies that <math>p \circ p^{-1} = \text{id}_{\mathbb{R}^2}</math> and <math>p^{-1} \circ p = \operatorname{id}_{S^2\setminus\{N\}}</math> by direct substitution. Hence <math>p</math> is a homeomorphism.<br />
}}<br />
<br />
=== Quotient space ===<br />
The unit interval <math>[0,1]</math> with the endpoints identified (the quotient space <math>[0,1]/\sim</math> where <math>0\sim 1</math>) is homeomorphic to the circle <math>S^1</math>.<br />
<br />
{{Proof|proof=Define the map <math>f:[0,1] \to S^1</math> by <math display="block">f(t)=(\cos(2\pi t), \sin(2\pi t)).</math> This map is continuous and [[Surjection|surjective]], and satisfies <math>f(0)=f(1)=(1,0)</math>.<br />
<br />
Consider the equivalence relation <math>\sim</math>, and let <math>q:[0,1]\to [0,1]/\sim</math> be the [[quotient map]]. By the [[universal property]] of the quotient map, there exists a unique continuous map <math>\tilde{f}: [0,1]/\sim \to S^1</math> such that <math>\tilde{f} \circ q = f</math>; that is, the following diagram commutes.<br />
<div style="text-align: center;"><br />
<br />
<kroki lang="tikz"><br />
\documentclass[tikz]{standalone}<br />
\usepackage{tikz-cd}<br />
\begin{document}<br />
\begin{tikzpicture}[baseline=(current bounding box.center)] <br />
\node[scale=1.5] {<br />
\begin{tikzcd}<br />
{[0,1]} && {S^1} \\<br />
& {[0,1]/{\sim}} \arrow["f", from=1-1, to=1-3]<br />
\arrow["q"', from=1-1, to=2-2]<br />
\arrow["{\exists! \tilde{f}}"', dashed, from=2-2, to=1-3]<br />
\end{tikzcd}<br />
};<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
The map <math>\tilde{f}</math> is bijective because:<br />
* Surjectivity follows from surjectivity of <math>f</math>;<br />
* [[Injection|Injectivity]] holds because <math display="block">\tilde{f}([t])=\tilde{f}([s])\Rightarrow t=s \text{ or } \{t,s\}=\{0,1\},</math> but in the latter case <math>[t]=[s]</math> in the quotient. <br />
<br />
Hence <math>\tilde{f}</math> is a continuous bijection.<br />
<br />
The space <math>[0,1]/\sim</math> is compact as the quotient of a [[compact space]], and <math>S^1</math> is [[Hausdorff space|Hausdorff]]. By the [[Compact-to-Hausdorff theorem]], a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.<br />
<br />
Therefore <math>\tilde{f}</math> is a homeomorphism.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz">\documentclass[tikz,border=10pt]{standalone}<br />
\usepackage{amsmath}<br />
\usetikzlibrary{arrows.meta,calc}<br />
\begin{document}<br />
<br />
\begin{tikzpicture}[<br />
>={Stealth[scale=1.1]},<br />
dot/.style={circle,fill=black,inner sep=1.6pt},<br />
label text/.style={font=\Large,align=center}<br />
]<br />
<br />
\def\r{1.4}<br />
\def\gap{50}<br />
<br />
\coordinate (C1) at (0,0);<br />
\coordinate (C2) at (5.5,0);<br />
\coordinate (C3) at (11,0);<br />
<br />
\begin{scope}[shift={(C1)}]<br />
\draw[thick] (-\r,0) coordinate (A) -- (\r,0) coordinate (B);<br />
\node[dot] at (A) {};<br />
\node[dot] at (B) {};<br />
\node[label text] at (0,-2.6) {$[0,1]$};<br />
\coordinate (R1) at (\r,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C1)+(2,0)$) -- ($(C2)+(-2,0)$)<br />
node[midway,above] {$q$};<br />
<br />
\begin{scope}[shift={(C2)}]<br />
\draw[thick]<br />
(180-\gap:\r)<br />
arc[start angle=180-\gap,end angle=360+\gap,radius=\r];<br />
<br />
\node[dot] (L) at (180-\gap:\r) {};<br />
\node[dot] (R) at (\gap:\r) {};<br />
<br />
\draw[dashed,->,thick]<br />
(L) .. controls +(0,0) and +(-0.8,-0.1) .. (90:\r);<br />
\draw[dashed,->,thick]<br />
(R) .. controls +(0,0) and +(0.8,-0.1) .. (90:\r);<br />
<br />
\node[label text] at (0,-2.6)<br />
{$[0,1]/\sim$ \\[-0.4ex]\normalsize $(0\sim1)$};<br />
<br />
\coordinate (R2) at (2,0);<br />
\end{scope}<br />
<br />
\draw[->,thick] ($(C2)+(2,0)$) -- ($(C3)+(-2,0)$)<br />
node[midway,above] {$\overset{\tilde{f}}{\cong}$};<br />
<br />
\begin{scope}[shift={(C3)}]<br />
\draw[thick] (0,0) circle (\r);<br />
\node[dot] at (90:\r) {};<br />
\node[label text] at (0,-2.6) {$S^{1}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}</kroki><br />
</div><br />
<br />
}}<br />
<br />
== Counterexamples ==<br />
<math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>.<br />
<br />
{{Proof|proof=For contradiction, suppose that there exists a homeomorphism <math>f: \mathbb{R}\to\mathbb{R}^2</math>.<br />
<br />
Consider the subspace <math>\mathbb{R}\setminus\{0\}</math> of <math>\mathbb{R}</math>. The [[restriction]] on it, <math>\left.f\right|_{\mathbb{R}\setminus\{0\}}: \mathbb{R}\setminus\{0\}\to \mathbb{R}^2\setminus\{f(0)\}</math> is also a homeomorphism.<br />
<br />
However, <math>\mathbb{R}\setminus\{0\}</math> has two connected components, <math>(-\infty,0)</math> and <math>(0,\infty)</math>, while <math>\mathbb{R}^2\setminus\{f(0)\}</math> is connected, which contradicts the assumption that the two spaces are homeomorphic.<br />
<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass[tikz, border=10pt]{standalone}<br />
\usepackage{amsmath, amssymb}<br />
<br />
\begin{document}<br />
\begin{tikzpicture}<br />
\begin{scope}[xshift=-5cm]<br />
\draw[thick] (-3, 0) -- (3, 0);<br />
<br />
\filldraw[fill=white, draw=black, thick] (0, 0) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2.5, 0) -- (-1, 0);<br />
\fill[cyan!60!blue] (-2.5, 0) circle (2pt);<br />
\fill[cyan!60!blue] (-1, 0) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (1.3, 0) -- (2.2, 0);<br />
\fill[cyan!60!blue] (1.3, 0) circle (2pt);<br />
\fill[cyan!60!blue] (2.2, 0) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -2) {$\mathbb{R} \setminus \{0\}$};<br />
\end{scope}<br />
<br />
\begin{scope}[xshift=4cm]<br />
\draw[thick] (-3.5, -2.5) -- (3.5, -2.5) -- (3.5, 2.5) -- (-3.5, 2.5) -- cycle;<br />
<br />
\filldraw[fill=white, draw=black, thick] (-0.3, -0.2) circle (2.5pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-2, 1.5) .. controls (-0.5, 1) and (-0.8, -1) .. (-1.2, -1.8);<br />
\fill[cyan!60!blue] (-2, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (-1.2, -1.8) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 1.5) .. controls (1.5, 1.8) and (2.5, 1) .. (2, 0.5);<br />
\fill[cyan!60!blue] (-0.3, 1.5) circle (2pt);<br />
\fill[cyan!60!blue] (2, 0.5) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.3, 0.5) .. controls (1, 0) and (1, -1) .. (2.5, -2);<br />
\fill[cyan!60!blue] (-0.3, 0.5) circle (2pt);<br />
\fill[cyan!60!blue] (2.5, -2) circle (2pt);<br />
<br />
\draw[cyan!60!blue, very thick] (-0.5, -0.8) .. controls (0, -1.8) and (1, -1.5) .. (0.8, -1);<br />
\fill[cyan!60!blue] (-0.5, -0.8) circle (2pt);<br />
\fill[cyan!60!blue] (0.8, -1) circle (2pt);<br />
<br />
\node[font=\Large] at (0, -3.5) {$\mathbb{R}^2 \setminus \{f(0)\}$};<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
\end{document}<br />
</kroki><br />
</div><br />
<br />
Hence, no such homeomorphism exists; therefore <math>\mathbb{R}</math> is not homeomorphic to <math>\mathbb{R}^2</math>}}The map from the interval <math>[0,1)</math> to the 1-sphere <math>S^1</math>,<br />
<math display="block">\phi: [0,1)\to S^1,\quad x\mapsto e^{2\pi ix}</math><br />
is continuous and bijective, but not a homeomorphism. <br />
{{Proof|proof=<br />
<div style="text-align: center;"><br />
<kroki lang="tikz"><br />
\documentclass{article}<br />
\usepackage{tikz}<br />
\usetikzlibrary{arrows.meta}<br />
<br />
\begin{document}<br />
<br />
\begin{tikzpicture}<br />
\draw (0,0) -- (4,0);<br />
<br />
\draw (0.15, 0.25) -- (0, 0.25) -- (0, -0.25) -- (0.15, -0.25);<br />
<br />
\draw (3.9, 0.25) to[bend left=45] (3.9, -0.25);<br />
<br />
\draw[-{Stealth[length=3mm, width=2mm]}] (4.5, 1.2) to[bend left=30] node[above=2pt] {$\phi$} (7.0, 1.2);<br />
<br />
\draw (9.5, 0) circle (2);<br />
<br />
\begin{scope}[rotate around={90:(11.5,0)}]<br />
\draw (11.39, 0.25) to[bend left=45] (11.39, -0.25);<br />
\draw (11.5, -0.25) -- (11.5, 0.25);<br />
\draw (11.5, 0.25) -- (11.65, 0.25);<br />
\draw (11.5, -0.25) -- (11.65, -0.25);<br />
<br />
\end{scope}<br />
<br />
\end{tikzpicture}<br />
<br />
\end{document}<br />
</kroki><br />
</div><br />
The map <math>\phi</math> is:<br />
* Continuous, as it is the composition of continuous maps <math>x\mapsto 2\pi x</math> and <math>t\mapsto e^{it}</math>.<br />
* Injective, because if <math>e^{2\pi i x_1}=e^{2\pi i x_2}</math>, then <math>x_1-x_2\in \mathbb{Z}</math>. Since <math>x_1,x_2\in [0,1)</math>, it follows that <math>x_1=x_2</math>.<br />
* Surjective, since every point of <math>S^1</math> can be written as <math>e^{2\pi i x}</math> for some <math>x\in [0,1)</math>.<br />
Hence <math>\phi</math> is a continuous bijection.<br />
<br />
However, <math>\phi</math> is not a homeomorphism.<br />
<br />
Consider the sequence<math display="block">z_n = e^{2\pi i (1-\tfrac{1}{n})} \in S^1.</math><br />
Then <math>z_n \to 1 = e^{2\pi i \cdot 0}</math> in <math>S^1</math>. But<br />
<math display="block">\phi^{-1}(z_n) = 1-\tfrac{1}{n} \to 1,</math><br />
which does not converge to <math>\phi^{-1}(1)=0</math> in <math>[0,1)</math>. Thus <math>\phi^{-1}</math> is not continuous.<br />
}}<br />
== Topological invariants ==<br />
A [[topological invariant]] is a property of a topological space that is preserved under homeomorphisms. In other words, if two spaces are homeomorphic, they either both possess the property or both do not. Invariants are the important tools to classify topological spaces. If two spaces differ in any topological invariant, they cannot be homeomorphic. Conversely, showing that two spaces share many invariants is often the first step on proving they are homeomorpic, though it is never sufficient by itself.<br />
<br />
=== Common topological invariants ===<br />
<br />
* [[Connectedness]]<br />
* [[Compactness]]<br />
* [[Hausdorff space|Hausdorff property]]<br />
* [[Cardinality]] of the space<br />
<br />
=== Algebraic invariants ===<br />
More powerful invariants come from [[algebraic topology]], which assigns algebraic objects to topological spaces.<br />
<br />
* [[Fundamental group]]<br />
* [[Homology group]]<br />
* [[Higher homotopy group]]<br />
<br />
==Homeomorphism group==<br />
The collection of all [[autohomeomorphisms]] of a topological space <math>X</math> forms a [[group]] under composition operation, known as the homeomorphism group of <math>X</math>, denoted <math>\operatorname{Homeo}(X)</math>. The homeomorphism group captures the symmetry in topology. It describes the ways in which a topological space can be continuously transformed onto itself.<br />
<br />
The homeomorphism group <math>\operatorname{Homeo}(X)</math> is a faithful [[group action]] on its underlying set <math>X</math>. It moves points in <math>X</math> continuously onto <math>X</math> itself, and the topological structure of <math>X</math> is also reflected in the algebraic invariants such as the [[Orbit|orbits]] and [[Stabilizer|stabilizers]] of the action.<br />
<br />
For example, consider the 2-sphere <math>S^2</math> as a thin rubber membrane tightly wraped around a ball. Each autohomeomorphism of <math>S^2</math>, which is an element in <math>\operatorname{Homeo}(S^2)</math>, corresponds to a continuous deformation of this membrane. This operation can be stretching, bending, twisting, or any composition of these operations, so the rubber always remains attached to the ball.<br />
<div style="text-align: center;"><br />
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\draw[cyan!70!black, thin] (0,2) -- (0,-2);<br />
<br />
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(0.7, 1.1) .. controls (1.0, 1.3) and (1.3, 1.1) .. (1.4, 0.5) <br />
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(0.05, 1.3) <br />
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</div><br />
Under the natural action of <math>\operatorname{Homeo}(S^2)</math>, every point on the sphere can be moved continuously to any other point. This example shows how the homeomorphism group captures the symmetry of a topological space in the perspective of continuity.<br />
<br />
==See also==<br />
* [[Homotopy]]<br />
* [[Topology]]<br />
* [[Homeomorphism group]]<br />
<br />
[[Category:Topopogy]]</div>InfernalAtom683