nLab
local homeomorphism

A local homeomorphism is a morphism $p:E\to B$ in the category Top of topological spaces and continuous maps betweem them such that

• for every $e\in E$, there is an open set $U\ni e$ such that the image $p_{*}(U)$ is open in $B$ and the restriction of $p$ to $U$ is a homeomorphism $p{\mid }_U:U\to p_{*}(U)$,

or equivalently

• for every $e\in E$, there is a neighbourhood $U$ of $e$ such that the image $p_{*}(U)$ is a neighbourhood of $p(e)$ and $p{\mid }_U:U\to p_{*}(U)$ is a homeomorphism.

See also etale space.