Homeomorphism
Not to be confused with Homomorphism
Contents
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:
- There exists a mapping [ilmath]f:(X,d)\rightarrow(Y,d')[/ilmath] such that:
- [ilmath]f[/ilmath] is bijective
- [ilmath]f[/ilmath] is continuous
- [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
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:
- [math]f[/math] is bijective
- [math]f[/math] is continuous
- [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[2]:
- [ilmath]f[/ilmath] be continuous
- [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:
- (Find ref for [ilmath]\cong[/ilmath])
- [ilmath]\approx[/ilmath][2] - NOTE: really rare, I've only ever seen this used to denote homeomorphism in this one book.
