# Homeomorphism

(Redirected from Homeomorphism (topology))
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.
The message provided is:
As a part of the topology patrol.

(Previous work dated 2nd May 2016)

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[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.

This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
For the [ilmath]\cong[/ilmath] notation - don't worry I haven't just made it up

## Notes

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.

# OLD PAGE

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 $f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})$ where:

1. $f$ is bijective
2. $f$ is continuous
3. $f^{-1}$ 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[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.