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 $𝑓 ↾ 𝑈 : 𝑈 \to 𝑓 (𝑈)$ is a homeomorphism. topology Equivalently, $𝑓$ is a local homeomorphism iff $𝑓$ is continuous, open, and locally injective.