# Étale map

## Definition

Let $X$ and $Y$ be topological spaces. A continuous map $f:X \to Y$ is termed an étale map if it is surjective, is a local homeomorphism, and if every fiber $f^{-1}(y)$ is discrete with the subspace topology.

## Relation with other properties

### Incomparable properties

• Bundle map (the map associated to a fiber bundle): A map which is both an etale map and a bundle map is a covering map