Local homeomorphism

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Definition

Let $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$ be topological spaces and let $f:X\rightarrow Y$ be a map (we do not require continuity at this stage). We call $f$ a local homeomorphism if:

• $\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 }$$\text{homeomorphism}$$\big)\big]$
  • In words: for all points $x\in X$ 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)

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