Étale map

This article defines a property of continuous maps between topological spaces

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

Stronger properties

• Covering map

Weaker properties

• Local homeomorphism
• Open map
• Quotient map

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