Local Homeomorphisms

Definition: Local Homeomorphism

Let $(X,{\tau }_X)$ and $(Y,{\tau }_Y)$ be Topological Spaces.

A local homeomorphism is a function $f:X\to Y$ such that every $x\in X$ has an open neighbourhood $O$ whose image $f(O)$ is open and for which the restriction $f{|}_O$ is a homeomorphism between the subspaces $O$ and $f(O)$.

Properties

Theorem

A local homeomorphism is a homeomorphism if and only if it is bijective.

PROOF

TODO