Difference between revisions of "Homeomorphism"
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Using [[Continuity definitions are equivalent]] it is easily seen that the metric space definition ''implies'' the topological definition. That is to say: | Using [[Continuity definitions are equivalent]] it is easily seen that the metric space definition ''implies'' the topological definition. That is to say: | ||
* If {{M|f}} is a (metric) homeomorphism then is is also a topological one (when the topologies considered are those [[Topology induced by a metric|those induced by the metric]]. | * If {{M|f}} is a (metric) homeomorphism then is is also a topological one (when the topologies considered are those [[Topology induced by a metric|those induced by the metric]]. | ||
| + | |||
| + | ==Terminology and notation== | ||
| + | If there exists a ''homeomorphism'' between two spaces, {{M|X}} and {{M|Y}} we say<ref name="FOAT"/>: | ||
| + | * {{M|X}} and {{M|Y}} are ''homeomorphic'' | ||
| + | |||
| + | The notations used (with ''most common first'') are: | ||
| + | # (Find ref for {{M|\cong}}) | ||
| + | # {{M|\approx}}<ref name="FOAT"/> - '''NOTE: ''' really rare, I've only ever seen this used to denote homeomorphism in this one book. | ||
==See also== | ==See also== | ||
* [[Composition of continuous maps is continuous]] | * [[Composition of continuous maps is continuous]] | ||
| + | * [[Diffeomorphism]] | ||
==References== | ==References== | ||
Revision as of 00:28, 9 October 2015
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.
