---
tag: MDH71ZZ8XX
kind: concept
name: Local homeomorphism
msc: [54C10, 54C05]
specializes: [0YCG648KDD]
sources:
  - url: https://en.wikipedia.org/wiki/Local_homeomorphism
same_as:
  wikidata: Q1052761
  wikipedia: Local homeomorphism
---

## Summary

A local homeomorphism is a map between [[topological spaces|0CQHFPGJ45]] that is a [[homeomorphism|0QT00PB5G2]] near each point, onto an open set.
It need not be injective or surjective as a whole.
Covering maps and the projections of étale spaces are local homeomorphisms, and every local homeomorphism is [[continuous|0VGFHTM75Y]] and [[open|0YCG648KDD]].

## Explanations

### L1 human

A local homeomorphism looks like a perfect copy when you only look at a small piece of the space.
Globally it may wrap the space around many times.
The map that winds the real line around the circle, sending $t$ to the point at angle $2\pi t$, is the standard example.
Each short interval goes onto an arc without any folding, but the whole line covers the circle infinitely often.