Difference between revisions of "Homeomorphism"

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Not to be confused with [[Homomorphism]]
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{{Refactor notice|grade=A|msg=As a part of the topology patrol.
  
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==Topological Homeomorphism==
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(Previous work dated 2nd May 2016)
  
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A topological homeomorphism is [[Bijection|bijective]] map between two [[Topological space|topological spaces]] <math>f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})</math> where:
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Things to add:
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* Given a bijective continuous map, say {{M|f:X\rightarrow Y}}, the following are equivalent{{rITTMJML}}:
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*# {{M|f}} is a homeomorphism
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*# {{M|f}} is an [[open map]]
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*# {{M|f}} is a [[closed map]]
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** Document this. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 22:56, 22 February 2017 (UTC)
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* [[Example:A bijective and continuous map that is not a homeomorphism]] [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 22:58, 22 February 2017 (UTC)
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}}
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: '''Note: ''' not to be confused with [[Homomorphism]] which is a [[Category Theory (subject)|categorical]] construct.
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__TOC__
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==Definition==
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If {{Top.|X|J}} and {{Top.|Y|K}} are [[topological space|topological spaces]] a ''homeomorphism from {{M|X}} to {{M|Y}}'' is a{{rITTMJML}}:
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* [[Bijective]] map, {{M|f:X\rightarrow Y}} where both {{M|f}} and {{M|f^{-1} }} (the [[inverse function]]) are [[continuous]]
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We may then say that {{M|X}} and {{M|Y}} (or {{Top.|X|J}} and {{Top.|Y|K}} if the topology isn't obvious) are ''homeomorphic''<ref name="ITTMJML"/> or ''topologically equivalent''<ref name="ITTMJML"/>, we write this as:
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* {{M|X\cong Y}} (or indeed {{M|(X,\mathcal{J})\cong(Y,\mathcal{K})}} if the topologies are not implicit)
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*: '''Note: ''' some authors<ref name="ITTMJML"/> use {{M|\approx}} instead of {{M|\cong}}<ref group="Note"> I recommend {{M|\cong}} although I admit it doesn't matter which you use ''as long as it isn't'' {{M|\simeq}} (which is typically used for [[isomorphism (category theory)|isomorphic spaces]]) as that notation is used almost universally for [[homotopy equivalence]]. I prefer {{M|\cong}} as {{M|\cong}} looks stronger than {{M|\simeq}}, and {{M|\approx}} 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.</ref> I recommend you use {{M|\cong}}.
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'''Claim 1:''' {{M|\cong}} is an [[equivalence relation]] on [[topological space|topological spaces]].
  
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[[Global topological properties]] are precisely those properties of [[topological space|topological spaces]] preserved by homeomorphism.
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{{Requires references|For the {{M|\cong}} notation - don't worry I haven't just made it up}}
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==Notes==
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<references group="Note"/>
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==References==
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<references/>
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=OLD PAGE=
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Not to be confused with [[Homomorphism]]
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==Homeomorphism of metric spaces==
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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:
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* There exists a [[Function|mapping]] {{M|f:(X,d)\rightarrow(Y,d')}} such that:
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*# {{M|f}} is [[Bijection|bijective]]
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*# {{M|f}} is [[Continuous map|continuous]]
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*# {{M|f^{-1} }} is also a [[Continuous map|continuous map]]
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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
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{{Todo|Find reference for use of {{M|\cong}} notation}}
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==Topological Homeomorphism==
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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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===Technicalities===
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{{Note|This section contains pedantry. The reader should be aware of it, but not concerned by not considering it}}
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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>:
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# {{M|f}} be continuous
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# {{M|f^{-1} }} exists and is continuous.
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===Agreement with metric definition===
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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:
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* 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]].
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==Terminology and notation==
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If there exists a ''homeomorphism'' between two spaces, {{M|X}} and {{M|Y}} we say<ref name="FOAT"/>:
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* {{M|X}} and {{M|Y}} are ''homeomorphic''
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The notations used (with ''most common first'') are:
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# (Find ref for {{M|\cong}})
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# {{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]]
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* [[Diffeomorphism]]
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==References==
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<references/>
  
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{{Definition|Topology}}
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{{Definition|Topology|Metric Space}}[[Category:Equivalence relations]]

Latest revision as of 22:58, 22 February 2017

Grade: A
This page is currently being refactored (along with many others)
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.
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As a part of the topology patrol.

(Previous work dated 2nd May 2016)

Things to add:

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.

(Unknown grade)
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.

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 Introduction to Topological Manifolds - John M. Lee

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 [math]f:(X,\mathcal{J})\rightarrow(Y,\mathcal{K})[/math] where:

  1. [math]f[/math] is bijective
  2. [math]f[/math] is continuous
  3. [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]:

  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.

See also

References

  1. ↑ Functional Analysis - George Bachman Lawrence Narici
  2. ↑ 2.0 2.1 2.2 Fundamentals of Algebraic Topology, Steven H. Weintraub
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