Difference between revisions of "Local homeomorphism"
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==Definition== | ==Definition== | ||
| − | Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[map]] (we do not require [[continuity]] at this stage). We call {{M|f}} a ''local homeomorphism'' if: | + | Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]] and let {{M|f:X\rightarrow Y}} be a [[map]] (we do not require [[continuity]] at this stage). We call {{M|f}} a ''local homeomorphism'' if{{rITTMJML}}: |
* {{M|\forall x\in X\exists U\in\mathcal{O}(x,X)\big[\big(f(U)\in\mathcal{K}\big)\wedge \big(f\vert_U^\text{Im}:U\rightarrow f(U)\text{ is a } }}[[homeomorphism|{{M|\text{homeomorphism} }}]]{{M|\big)\big]}}<ref group="Note">Note about notation: | * {{M|\forall x\in X\exists U\in\mathcal{O}(x,X)\big[\big(f(U)\in\mathcal{K}\big)\wedge \big(f\vert_U^\text{Im}:U\rightarrow f(U)\text{ is a } }}[[homeomorphism|{{M|\text{homeomorphism} }}]]{{M|\big)\big]}}<ref group="Note">Note about notation: | ||
* {{M|f\vert_A^\text{Im}:A\rightarrow f(A)}} is the [[restriction onto its image]] of a [[function]]. | * {{M|f\vert_A^\text{Im}:A\rightarrow f(A)}} is the [[restriction onto its image]] of a [[function]]. | ||
* {{M|\mathcal{O}(x,X)}} is the [[set of open neighbourhoods of a point in a topological space]]</ref> | * {{M|\mathcal{O}(x,X)}} is the [[set of open neighbourhoods of a point in a topological space]]</ref> | ||
** In words: for all points {{M|x\in X}} there exists [[open neighbourhood|open neighbourhoods]] of {{M|x}}, say {{M|U}}, that {{M|f(U)}} is open in {{M|Y}} and {{M|f}} restricted to {{M|U}} (onto the image of {{M|U}}) is a [[homeomorphism]] (when {{M|U}} and {{M|f(U)}} are considered with the [[subspace topology]] of course) | ** In words: for all points {{M|x\in X}} there exists [[open neighbourhood|open neighbourhoods]] of {{M|x}}, say {{M|U}}, that {{M|f(U)}} is open in {{M|Y}} and {{M|f}} restricted to {{M|U}} (onto the image of {{M|U}}) is a [[homeomorphism]] (when {{M|U}} and {{M|f(U)}} are considered with the [[subspace topology]] of course) | ||
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| + | If there is a ''local'' homeomorphism between two spaces we say they are ''locally homeomorphic'' | ||
==Immediate properties== | ==Immediate properties== | ||
| + | {{XXX|I do not know if local homeomorphism is preserved by anything, or an equivalence relation}} - investigate this. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:45, 22 February 2017 (UTC) | ||
* [[A local homeomorphism is continuous]] | * [[A local homeomorphism is continuous]] | ||
* [[A local homeomorphism is an open map]] | * [[A local homeomorphism is an open map]] | ||
Latest revision as of 21:45, 22 February 2017
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Prototype page
Contents
[hide]Definition
Let (X,J) and (Y,K) be topological spaces and let f:X→Y be a map (we do not require continuity at this stage). We call f a local homeomorphism if[1]:
- ∀x∈X∃U∈O(x,X)[(f(U)∈K)∧(f|ImU:U→f(U) is a homeomorphism)][Note 1]
- In words: for all points x∈X there exists open neighbourhoods of x, say U, that f(U) is open in Y and f restricted to U (onto the image of U) is a homeomorphism (when U and f(U) are considered with the subspace topology of course)
If there is a local homeomorphism between two spaces we say they are locally homeomorphic
Immediate properties
TODO: I do not know if local homeomorphism is preserved by anything, or an equivalence relation
- investigate this. Alec (talk) 21:45, 22 February 2017 (UTC)
- A local homeomorphism is continuous
- A local homeomorphism is an open map
- A bijective local homeomorphism is a homeomorphism
- Every homeomorphism is a local homeomorphism
Notes
- Jump up ↑ Note about notation:
- f|ImA:A→f(A) is the restriction onto its image of a function.
- O(x,X) is the set of open neighbourhoods of a point in a topological space
