Let \(X\) and \(Y\) be topological spaces. The space \(X\) is locally homeomorphic to \(Y\) if every \(x\in X\) has an open neighborhood \(U_x\) that is homeomorphic to an open subset of \(Y\).
If there is a local homeomorphism \(f\colon X\to Y\), then \(X\) is locally homeomorphic to \(Y\); the converse fails.
The sphere \(S^2\) is locally homeomorphic to \(\RR^2\), by stereographic projection from each of two antipodal points, but there is no local homeomorphism \(f\colon S^2 \to \RR^2\): the image \(f(S^2)\) would be a nonempty subset of \(\RR^2\) that is open, because local homeomorphisms are open maps, and closed, because it is compact, hence equal to the noncompact space \(\RR^2\).
What does it mean for $X$ to be locally homeomorphic to $Y$?