A function \(f\colon X\to Y\) between two topological spaces is called local homeomorphism if for every point \(x\in X\) there is a neighbourhood \(U\subseteq X\) such that \(f(U)\) is open in \(Y\) and the restriction \(f|_U\colon U\to f(U)\) is a homeomorphism .
Remarks
- Every homeomorphism is a local homeomorphism.
- Every local homeomorphism is continuous and open.
- Every bijective local homeomorphism is a homeomorphism.