Local homeomorphism

Definition
Let $$X$$ and $$Y$$ be topological spaces. A continuous map $$f:X \to Y$$ is termed a local homeomorphism if the following are true:


 * It is an open map
 * Every $$x \in X$$ has an open neighbourhood $$U$$ such that $$f|_U$$ is a homeomorphism to its image and $$f(U)$$ is itself an open subset of $$Y$$.

Some variants of the definition of local homeomorphism also require the map to be surjective.