Difference between revisions of "Homeomorphism"
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Not to be confused with [[Homomorphism]] | Not to be confused with [[Homomorphism]] | ||
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| + | ==Homeomorphism of metric spaces== | ||
| + | Given two [[Metric space|metric spaces]] {{M|(X,d)}} and {{M|(Y,d')}} they are said to be ''homeomorphic''<ref name="FA">Functional Analysis - George Bachman Lawrence Narici</ref> if: | ||
| + | * There exists a [[Function|mapping]] {{M|f:(X,d)\rightarrow(Y,d')}} such that: | ||
| + | *# {{M|f}} is [[Bijection|bijective]] | ||
| + | *# {{M|f}} is [[Continuous map|continuous]] | ||
| + | *# {{M|f^{-1} }} is also a [[Continuous map|continuous map]] | ||
| + | Then {{M|(X,d)}} and {{M|(Y,d')}} are ''homeomorphic'' and we may write {{M|(X,d)\cong(Y,d')}} or simply (as [[Mathematicians are lazy]]) {{M|X\cong Y}} if the metrics are obvious | ||
| + | {{Todo|Find reference for use of {{M|\cong}} notation}} | ||
==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]] | ||
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| + | {{Todo|Using [[Continuity definitions are equivalent]] it is easily seen that the metric space definition implies the second, that logic and a reference would be good!}} | ||
==See also== | ==See also== | ||
* [[Composition of continuous maps is continuous]] | * [[Composition of continuous maps is continuous]] | ||
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| + | ==References== | ||
| + | <references/> | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 13:27, 9 July 2015
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:
- 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
TODO: Using Continuity definitions are equivalent it is easily seen that the metric space definition implies the second, that logic and a reference would be good!
See also
References
- ↑ Functional Analysis - George Bachman Lawrence Narici
