Local homeomorphism

A mapping $f : X \rightarrow Y$ between topological spaces such that for every point $x \in X$ there is a neighbourhood $\mathcal{O}_x$ that maps homeomorphically into $Y$ under $f$ (cf. Homeomorphism). Sometimes in the definition of a local homeomorphism the requirement $f(X) = Y$ is included and $f$ is also assumed to be an open mapping. Examples of local homeomorphisms are: a continuously-differentiable mapping with non-zero Jacobian on an open subset of an $n$-dimensional Euclidean space into the $n$-dimensional Euclidean space; a covering mapping, in particular the natural mapping of a topological group onto its quotient space with respect to a discrete subgroup. If the mapping $f : X \rightarrow Y$ of a Čech-complete space, in particular a locally compact Hausdorff space, onto a Tikhonov space $Y$ is open and countable-to-one, that is, $|f^{-1}(y)| \le \aleph_0$, $y \in Y$, then on some open everywhere-dense set in $X$ the mapping $f$ is a local homeomorphism.

References

[a1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)