[[Topology MOC]]
# Local homeomorphism

A **local homeomorphism** is a map $f : X \to Y$ between topological spaces
such that every $x \in X$ has a neighbourhood $U$ such that $f(U)$ is open and $f \restriction U : U \to f(U)$ is a [[homeomorphism]]. #m/def/topology 
Equivalently, $f$ is a local homeomorphism iff $f$ is [[Continuity|continuous]], [[Open and closed maps|open]], and [[Local injection|locally injective]].

> [!missing]- Proof
> #missing/proof

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