Local homeomorphism
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Definition
Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces and let [ilmath]f:X\rightarrow Y[/ilmath] be a map (we do not require continuity at this stage). We call [ilmath]f[/ilmath] a local homeomorphism if:
- [ilmath]\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 } [/ilmath][ilmath]\text{homeomorphism} [/ilmath][ilmath]\big)\big][/ilmath][Note 1]
- In words: for all points [ilmath]x\in X[/ilmath] there exists open neighbourhoods of [ilmath]x[/ilmath], say [ilmath]U[/ilmath], that [ilmath]f(U)[/ilmath] is open in [ilmath]Y[/ilmath] and [ilmath]f[/ilmath] restricted to [ilmath]U[/ilmath] (onto the image of [ilmath]U[/ilmath]) is a homeomorphism (when [ilmath]U[/ilmath] and [ilmath]f(U)[/ilmath] are considered with the subspace topology of course)
Immediate properties
- 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
- ↑ Note about notation:
- [ilmath]f\vert_A^\text{Im}:A\rightarrow f(A)[/ilmath] is the restriction onto its image of a function.
- [ilmath]\mathcal{O}(x,X)[/ilmath] is the set of open neighbourhoods of a point in a topological space
