Let \(X\) and \(Y\) be topological spaces. A continuous map \(f\colon X\to Y\) is a local homeomorphism if every \(x\in X\) has an open neighborhood \(U_x\) such that \(f(U_x)\) is open in \(Y\) and \(f\vert_{U_x}\colon U_x \to f(U_x)\) is a homeomorphism.
Every covering space \(p\colon\tilde X\to X\) is a local homeomorphism. For a sheaf of sets \(\mathcal F\) on a space \(X\), the projection from its étale space to \(X\) is a local homeomorphism.
When is a map $f: X \to Y$ a local homeomorphism?