Homeomorphism: Difference between revisions
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== Examples == | == Examples == | ||
=== Open interval | === Open interval === | ||
The [[open interval]] <math>(0,1)</math> is homeomorphic to <math>\mathbb{R}</math>. | 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 | {{Proof|proof=The map <math>f:(0,1)\to \mathbb{R}</math> defined by | ||
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is continuous. Thus <math>f</math> is a homeomorphism.}} | is continuous. Thus <math>f</math> is a homeomorphism.}} | ||
=== | === Stereographic projection === | ||
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]]. | 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]]. | ||
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}} | }} | ||
=== | === Quotient space === | ||
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>. | 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> | {{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. | 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. | ||
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== Counterexamples == | == Counterexamples == | ||
<math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>. | <math>\mathbb{R}</math> is '''not''' homeomorphic to <math>\mathbb{R}^2</math>. | ||
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</div> | </div> | ||
Hence, no such homeomorphism exists; therefore <math>\mathbb{R}</math> is not homeomorphic to <math>\mathbb{R}^2</math>}} | 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>, | ||
The map from the interval <math>[0,1)</math> to the 1-sphere <math>S^1</math>, | |||
<math display="block">\phi: [0,1)\to S^1,\quad x\mapsto e^{2\pi ix}</math> | <math display="block">\phi: [0,1)\to S^1,\quad x\mapsto e^{2\pi ix}</math> | ||
is continuous and bijective, but not a homeomorphism. | is continuous and bijective, but not a homeomorphism. | ||
Revision as of 15:42, 1 April 2026
A homeomorphism is a special type of function between two topological spaces, that establishes that the two spaces are fundamentally the same from a topological perspective. Specifically, it is a continuous bijective function whose inverse function is also continuous. Homeomorphisms are the isomorphisms in the category of topological spaces 
, 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 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 
between topological spaces 
and 
is called a homeomorphism, if:
- is continuous,
- is bijective,
is continuous.
Two topological spaces and are called homeomorphic if there exists a homeomorphism between them, denoted 
.
Equivalent Definitions
A homeomorphism is a bijection that is continuous and open, or continuous and closed.
Properties
The composition of two homeomorphisms is again a homeomorphism.
Let 
and 
be homeomorphisms. Then:
is bijective, since the composition of two bijections is a bijection.
is continuous, as the composition of two continuous functions.
- The inverse is
, which is continuous because it is the composition of the continuous functions 
and .
Thus satisfies all requirements of a homeomorphism.
The inverse of a homeomorphism is again a homeomorphism.
Let be a homeomorphism. Then:
- is continuous by definition,
- is bijective, since the inverse of a bijection is again a bijection,
is continuous by definition.
Homeomorphism is an equivalence relation.
- Reflexivity: The identity map
is a continuous bijection on any topological space , whose inverse is itself. Thus 
is a homeomorphism. - Symmetry: If
is a homeomorphism, then its inverse 
is again a homeomorphism. - Transitivity: If and
are homeomorphisms, then 
is again a homeomorphism.
Examples
Open interval
The open interval 
is homeomorphic to 
.
The map 
defined by
is continuous because it is a composition of continuous functions. The restriction 
is bijective with continuous inverse 
. Therefore is bijective and its inverse
is a homeomorphism.
Stereographic projection
The Euclidean plane 
is homeomorphic to the 2-sphere minus one point, denoted 
where 
is the north pole.
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Define the stereographic projection 
by
on 
).
The inverse map 
is given by
and 
by direct substitution. Hence 
is a homeomorphism.
Quotient space
The unit interval 
with the endpoints identified (the quotient space 
where 
) is homeomorphic to the circle 
.
Define the map 
by
.
Consider the equivalence relation 
, and let 
be the quotient map. By the universal property of the quotient map, there exists a unique continuous map 
such that 
; that is, the following diagram commutes.
The map 
is bijective because:
- Surjectivity follows from surjectivity of ;
- Injectivity holds because
but in the latter case
in the quotient.
Hence is a continuous bijection.
The space is compact as the quotient of a compact space, and is Hausdorff. By the Compact-to-Hausdorff theorem, a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
Therefore is a homeomorphism.
HTTP-Response:
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l.23 \node[label text] at (0,-2.6) {^^?
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\end{document}Counterexamples
is not homeomorphic to .
For contradiction, suppose that there exists a homeomorphism 
.
Consider the subspace 
of . The restriction on it, 
is also a homeomorphism.
However, has two connected components, 
and 
, while 
is connected, which contradicts the assumption that the two spaces are homeomorphic.
HTTP-Response:
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Hence, no such homeomorphism exists; therefore is not homeomorphic to
The map from the interval 
to the 1-sphere ,
is continuous and bijective, but not a homeomorphism.
HTTP-Response:
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The map is:
- Continuous, as it is the composition of continuous maps
and 
. - Injective, because if
, then 
. Since 
, it follows that 
. - Surjective, since every point of can be written as
for some 
.
Hence is a continuous bijection.
However, is not a homeomorphism.
Consider the sequence
in . But
in . Thus 
is not continuous.
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 property
- Cardinality of the space
Algebraic invariants
More powerful invariants come from algebraic topology, which assigns algebraic objects to topological spaces.
Homeomorphism group
The collection of all autohomeomorphisms of a topological space forms a group under composition operation, known as the homeomorphism group of , denoted 
. 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 is a faithful group action on its underlying set . It moves points in continuously onto itself, and the topological structure of is also reflected in the algebraic invariants such as the orbits and stabilizers of the action.
For example, consider the 2-sphere 
as a thin rubber membrane tightly wraped around a ball. Each autohomeomorphism of , which is an element in 
, 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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Under the natural action of , 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.