View source for Homeomorphism

[[File:Topology joke.jpg|thumb|250x250px|A homeomorphism that turns a coffee mug into a donut continuously.]]
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'''. 

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.

== Definitions ==
A function <math>f</math> between topological spaces <math>X</math> and <math>Y</math> is called a '''homeomorphism''', if:

* <math>f</math> is continuous,
* <math>f</math> is bijective,
* <math>f^{-1}</math> is continuous.

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>.

=== Equivalent Definitions ===
A homeomorphism is a bijection that is continuous and [[Open function|open]], or continuous and [[Closed function|closed]].

== Properties ==
{{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:

* <math>g \circ f: X \to Z</math> is bijective, since the composition of two bijections is a bijection.

* <math>g \circ f</math> is continuous, as the composition of two continuous functions.

* 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>.

Thus <math>g \circ f</math> satisfies all requirements of a homeomorphism.}}


{{Property|property=The inverse of a homeomorphism is again a homeomorphism.}}
{{Proof|proof=Let <math>f: X \to Y</math> be a homeomorphism. Then:
* <math>f^{-1}</math> is continuous by definition,
* <math>f^{-1}</math> is bijective, since the inverse of a bijection is again a bijection,
* <math>\left(f^{-1}\right)^{-1}=f</math> is continuous by definition.}}


{{Property|property=Homeomorphism is an [[equivalence relation]].}}

{{Proof|proof=
* '''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.
* '''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.
* '''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.
}}

== Examples ==
The [[open interval]] <math>(0,1)</math> is homeomorphic to <math>\mathbb{R}</math>.
{{Proof|proof=The map <math>f:(0,1)\to \mathbb{R}</math> defined by
<math display="block">f(x)=\tan\left(\pi\left(x-\dfrac12\right)\right)</math>
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
<math display="block">f^{-1}(y)=\dfrac1\pi\arctan(y)+\dfrac12</math>
is continuous. Thus <math>f</math> is a homeomorphism.}}

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]].

{{Proof|proof=
<div style="float:right; width:320px; margin:0 0 0.5em 1em;">
<kroki lang="tikz">
\documentclass[tikz,border=15pt]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{calc, arrows.meta}
\begin{document}
\def\viewTheta{70}
\def\viewPhi{20}
\tdplotsetmaincoords{\viewTheta}{\viewPhi}

\begin{tikzpicture}[tdplot_main_coords, scale=2, line cap=round, line join=round]

    \def\R{1}
    \coordinate (O) at (0,0,0);
    \coordinate (N) at (0,0,\R);

    \def\thetaS{60}
    \def\phiS{30}
    \pgfmathsetmacro{\px}{\R * sin(\thetaS) * cos(\phiS)}
    \pgfmathsetmacro{\py}{\R * sin(\thetaS) * sin(\phiS)}
    \pgfmathsetmacro{\pz}{\R * cos(\thetaS)}
    \coordinate (P) at (\px, \py, \pz);

    \pgfmathsetmacro{\ux}{\px / (1 - \pz)}
    \pgfmathsetmacro{\uy}{\py / (1 - \pz)}
    \coordinate (Pprime) at (\ux, \uy, 0);

    \pgfmathsetmacro{\eqFrontStart}{\viewPhi - 180}
    \pgfmathsetmacro{\eqFrontEnd}{\viewPhi}
    \pgfmathsetmacro{\eqBackStart}{\viewPhi - 180}
    \pgfmathsetmacro{\eqBackEnd}{\viewPhi+180}

    \pgfmathsetmacro{\cotViewTheta}{cos(\viewTheta)/sin(\viewTheta)}
    \pgfmathsetmacro{\cotThetaS}{cos(\thetaS)/sin(\thetaS)}
    \pgfmathsetmacro{\cosAlpha}{max(min(-\cotThetaS * \cotViewTheta, 1), -1)}
    \pgfmathsetmacro{\alpha}{acos(\cosAlpha)}
    
    \pgfmathsetmacro{\latFrontStart}{\viewPhi-180}
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    \draw[thick, dashed] (0,0,0) -- (N);
    \draw[thick, ->, >=Stealth] (N) -- (0,0,1.8) node[anchor=south]{$z$};
    \begin{scope}[tdplot_screen_coords]
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    \tdplotdrawarc[cyan!60!blue, thick]{(O)}{\R}{\eqFrontStart}{\eqFrontEnd}{}{}
    \tdplotdrawarc[cyan!60!blue, thin, dashed]{(O)}{\R}{\eqBackStart}{\eqBackEnd}{}{}

    \pgfmathsetmacro{\rLat}{\R * sin(\thetaS)}
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    \tdplotsetrotatedcoordsorigin{(O)}
    
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    \draw[red, thick, ->, >=Stealth] (P) -- (Pprime);

    \fill[black] (N) circle (0.8pt) node[anchor=south east] {$N$};
    \fill[red] (P) circle (1pt) node[anchor=south west, text=black] {$(x,y,z)$};
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\end{tikzpicture}

\end{document}
</kroki>
</div>
Define the [[stereographic projection]] <math>p: S^2 \setminus \{N\} \to \mathbb{R}^2</math> by
<math display="block">p(x,y,z) = \left( \frac{x}{1-z}, \frac{y}{1-z} \right).</math>
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>).

The inverse map <math>p^{-1}: \mathbb{R}^2 \to S^2 \setminus \{N\}</math> is given by
<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>
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.
}}

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>.

{{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>.

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.
<div style="text-align: center;">

<kroki lang="tikz">
\documentclass[tikz]{standalone}
\usepackage{tikz-cd}
\begin{document}
\begin{tikzpicture}[baseline=(current bounding box.center)]  
\node[scale=1.5] {
    \begin{tikzcd}
	{[0,1]} && {S^1} \\
	& {[0,1]/{\sim}} \arrow["f", from=1-1, to=1-3]
	\arrow["q"', from=1-1, to=2-2]
	\arrow["{\exists! \tilde{f}}"', dashed, from=2-2, to=1-3]
    \end{tikzcd}
  };
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\end{document}
</kroki>
</div>

The map <math>\tilde{f}</math> is bijective because:
* Surjectivity follows from surjectivity of <math>f</math>;
* [[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. 

Hence <math>\tilde{f}</math> is a continuous bijection.

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.

Therefore <math>\tilde{f}</math> is a homeomorphism.

<div style="text-align: center;">
<kroki lang="tikz">\documentclass[tikz,border=10pt]{standalone}
\usepackage{amsmath}
\usetikzlibrary{arrows.meta,calc}
\begin{document}

\begin{tikzpicture}[
>={Stealth[scale=1.1]},
dot/.style={circle,fill=black,inner sep=1.6pt},
label text/.style={font=\Large,align=center}
]

\def\r{1.4}
\def\gap{50}

\coordinate (C1) at (0,0);
\coordinate (C2) at (5.5,0);
\coordinate (C3) at (11,0);

\begin{scope}[shift={(C1)}]
\draw[thick] (-\r,0) coordinate (A) -- (\r,0) coordinate (B);
\node[dot] at (A) {};
\node[dot] at (B) {};
\node[label text] at (0,-2.6) {$[0,1]$};
\coordinate (R1) at (\r,0);
\end{scope}

\draw[->,thick] ($(C1)+(2,0)$) -- ($(C2)+(-2,0)$)
node[midway,above] {$q$};

\begin{scope}[shift={(C2)}]
\draw[thick]
(180-\gap:\r)
arc[start angle=180-\gap,end angle=360+\gap,radius=\r];

\node[dot] (L) at (180-\gap:\r) {};
\node[dot] (R) at (\gap:\r) {};

\draw[dashed,->,thick]
(L) .. controls +(0,0) and +(-0.8,-0.1) .. (90:\r);
\draw[dashed,->,thick]
(R) .. controls +(0,0) and +(0.8,-0.1) .. (90:\r);

\node[label text] at (0,-2.6)
{$[0,1]/\sim$ \\[-0.4ex]\normalsize $(0\sim1)$};

\coordinate (R2) at (2,0);
\end{scope}

\draw[->,thick] ($(C2)+(2,0)$) -- ($(C3)+(-2,0)$)
node[midway,above] {$\overset{\tilde{f}}{\cong}$};

\begin{scope}[shift={(C3)}]
\draw[thick] (0,0) circle (\r);
\node[dot] at (90:\r) {};
\node[label text] at (0,-2.6) {$S^{1}$};
\end{scope}

\end{tikzpicture}
\end{document}</kroki>
</div>

}}

== Counterexamples ==
<math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>.

{{Proof|proof=For contradiction, suppose that there exists a homeomorphism <math>f: \mathbb{R}\to\mathbb{R}^2</math>.

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.

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.

<div style="text-align: center;">
<kroki lang="tikz">
\documentclass[tikz, border=10pt]{standalone}
\usepackage{amsmath, amssymb}

\begin{document}
\begin{tikzpicture}
    \begin{scope}[xshift=-5cm]
        \draw[thick] (-3, 0) -- (3, 0);
        
        \filldraw[fill=white, draw=black, thick] (0, 0) circle (2.5pt);
        
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        \draw[cyan!60!blue, very thick] (1.3, 0) -- (2.2, 0);
        \fill[cyan!60!blue] (1.3, 0) circle (2pt);
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        \node[font=\Large] at (0, -2) {$\mathbb{R} \setminus \{0\}$};
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    \begin{scope}[xshift=4cm]
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        \draw[cyan!60!blue, very thick] (-0.3, 1.5) .. controls (1.5, 1.8) and (2.5, 1) .. (2, 0.5);
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        \draw[cyan!60!blue, very thick] (-0.3, 0.5) .. controls (1, 0) and (1, -1) .. (2.5, -2);
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        \fill[cyan!60!blue] (2.5, -2) circle (2pt);
        
        \draw[cyan!60!blue, very thick] (-0.5, -0.8) .. controls (0, -1.8) and (1, -1.5) .. (0.8, -1);
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\end{tikzpicture}
\end{document}
</kroki>
</div>

Hence, no such homeomorphism exists; therefore <math>\mathbb{R}</math> is not homeomorphic to <math>\mathbb{R}^2</math>}}

== Topological invariants ==
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.

=== Common topological invariants ===

* [[Connectedness]]
* [[Compactness]]
* [[Hausdorff space|Hausdorff property]]
* [[Cardinality]] of the space

=== Algebraic invariants ===
More powerful invariants come from [[algebraic topology]], which assigns algebraic objects to topological spaces.

* [[Fundamental group]]
* [[Homology group]]
* [[Higher homotopy group]]

==Homeomorphism group==
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.

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.

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.
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        \filldraw[black] (0.8, 0.6) circle (1pt) node[anchor=north west, scale=0.8] {$\phi(y)$};
        
        \node[below, text=black] at (0,-2.3) {$\phi\cdot X$};
        \draw[cyan!60!blue, thick] (0,0) circle (2);
    \end{scope}
\end{tikzpicture}
\end{document}
</kroki>
</div>
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.

==See also==
* [[Homotopy]]
* [[Topology]]
* [[Homeomorphism group]]

[[Category:Topopogy]]