Homeomorphism

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As a part of the topology patrol

Note: not to be confused with Homomorphism which is a categorical construct.

Definition

If [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] are topological spaces a homeomorphism from [ilmath]X[/ilmath] to [ilmath]Y[/ilmath] is a:

• Bijective map, [ilmath]f:X\rightarrow Y[/ilmath] where both [ilmath]f[/ilmath] and [ilmath]f^{-1} [/ilmath] (the inverse function) are continuous

We may then say that [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] (or [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] if the topology isn't obvious) are homeomorphic or topologically equivalent, we write this as:

• [ilmath]X\cong Y[/ilmath] (or indeed [ilmath](X,\mathcal{J})\cong(Y,\mathcal{K})[/ilmath] if the topologies are not implicit)

  Note: some authors use [ilmath]\approx[/ilmath] instead of [ilmath]\cong[/ilmath] I recommend you use [ilmath]\cong[/ilmath].

Claim 1: [ilmath]\cong[/ilmath] is an equivalence relation on topological spaces.

Global topological properties are precisely those properties of topological spaces preserved by homeomorphism.

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For the [ilmath]\cong[/ilmath] notation - don't worry I haven't just made it up

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Not to be confused with Homomorphism

Homeomorphism of metric spaces

Given two metric spaces [ilmath](X,d)[/ilmath] and [ilmath](Y,d')[/ilmath] they are said to be homeomorphic if:

• There exists a mapping [ilmath]f:(X,d)\rightarrow(Y,d')[/ilmath] such that:
  1. [ilmath]f[/ilmath] is bijective
  2. [ilmath]f[/ilmath] is continuous
  3. [ilmath]f^{-1} [/ilmath] is also a continuous map

Then [ilmath](X,d)[/ilmath] and [ilmath](Y,d')[/ilmath] are homeomorphic and we may write [ilmath](X,d)\cong(Y,d')[/ilmath] or simply (as Mathematicians are lazy) [ilmath]X\cong Y[/ilmath] if the metrics are obvious

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TODO: Find reference for use of [ilmath]\cong[/ilmath] notation

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Topological Homeomorphism

A topological homeomorphism is bijective map between two topological spaces [math]f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})[/math] where:

1. [math]f[/math] is bijective
2. [math]f[/math] is continuous
3. [math]f^{-1}[/math] is continuous

Technicalities

This section contains pedantry. The reader should be aware of it, but not concerned by not considering it In order for [ilmath]f^{-1} [/ilmath] to exist, [ilmath]f[/ilmath] must be bijective. So the definition need only require:

1. [ilmath]f[/ilmath] be continuous
2. [ilmath]f^{-1} [/ilmath] exists and is continuous.

Agreement with metric definition

Using Continuity definitions are equivalent it is easily seen that the metric space definition implies the topological definition. That is to say:

• If [ilmath]f[/ilmath] is a (metric) homeomorphism then is is also a topological one (when the topologies considered are those those induced by the metric.

Terminology and notation

If there exists a homeomorphism between two spaces, [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] we say:

• [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] are homeomorphic

The notations used (with most common first) are:

1. (Find ref for [ilmath]\cong[/ilmath])
2. [ilmath]\approx[/ilmath] - NOTE: really rare, I've only ever seen this used to denote homeomorphism in this one book.

See also

• Composition of continuous maps is continuous
• Diffeomorphism