Difference between revisions of "Homeomorphism"
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==Topological Homeomorphism== | ==Topological Homeomorphism== | ||
A ''topological homeomorphism'' is [[Bijection|bijective]] map between two [[Topological space|topological spaces]] <math>f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})</math> where: | A ''topological homeomorphism'' is [[Bijection|bijective]] map between two [[Topological space|topological spaces]] <math>f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})</math> where: | ||
| − | |||
# <math>f</math> is [[Bijection|bijective]] | # <math>f</math> is [[Bijection|bijective]] | ||
# <math>f</math> is [[Continuous map|continuous]] | # <math>f</math> is [[Continuous map|continuous]] | ||
# <math>f^{-1}</math> is [[Continuous map|continuous]] | # <math>f^{-1}</math> is [[Continuous map|continuous]] | ||
| − | + | ===Technicalities=== | |
| − | {{ | + | {{Note|This section contains pedantry. The reader should be aware of it, but not concerned by not considering it}} |
| + | In order for {{M|f^{-1} }} to exist, {{M|f}} must be [[Bijection|bijective]]. So the definition need only require<ref name="FOAT">Fundamentals of Algebraic Topology, Steven H. Weintraub</ref>: | ||
| + | # {{M|f}} be continuous | ||
| + | # {{M|f^{-1} }} 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 {{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]]. | ||
==See also== | ==See also== | ||
Revision as of 00:22, 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.
