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As a part of the topology patrol.

(Previous work dated 2nd May 2016)

Things to add:

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


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[1]:

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[1] or topologically equivalent[1], 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[1] use [ilmath]\approx[/ilmath] instead of [ilmath]\cong[/ilmath][Note 1] 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


  1. I recommend [ilmath]\cong[/ilmath] although I admit it doesn't matter which you use as long as it isn't [ilmath]\simeq[/ilmath] (which is typically used for isomorphic spaces) as that notation is used almost universally for homotopy equivalence. I prefer [ilmath]\cong[/ilmath] as [ilmath]\cong[/ilmath] looks stronger than [ilmath]\simeq[/ilmath], and [ilmath]\approx[/ilmath] is the symbol for approximation, there is no approximation here. If you have a bijection, and both directions are continuous, the spaces are in no real way distinguishable.


  1. 1.0 1.1 1.2 1.3 1.4 Introduction to Topological Manifolds - John M. Lee


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[1] if:

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

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

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


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[2]:

  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[2]:

  • [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][2] - NOTE: really rare, I've only ever seen this used to denote homeomorphism in this one book.

See also


  1. Functional Analysis - George Bachman Lawrence Narici
  2. 2.0 2.1 2.2 Fundamentals of Algebraic Topology, Steven H. Weintraub