Homeomorphism
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(Previous work dated 2nd May 2016)
Things to add:
- Given a bijective continuous map, say [ilmath]f:X\rightarrow Y[/ilmath], the following are equivalent^{[1]}:
- [ilmath]f[/ilmath] is a homeomorphism
- [ilmath]f[/ilmath] is an open map
- [ilmath]f[/ilmath] is a closed map
- Example:A bijective and continuous map that is not a homeomorphism Alec (talk) 22:58, 22 February 2017 (UTC)
- Note: not to be confused with Homomorphism which is a categorical construct.
Contents
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]}:
- Bijective map, [ilmath]f:X\rightarrow Y[/ilmath] where both [ilmath]f[/ilmath] and [ilmath]f^{-1} [/ilmath] (the inverse function) are continuous
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.
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Notes
- ↑ 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
- ↑ ^{1.0} ^{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:
- 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.
See also
References
- ↑ Functional Analysis - George Bachman Lawrence Narici
- ↑ ^{2.0} ^{2.1} ^{2.2} Fundamentals of Algebraic Topology, Steven H. Weintraub