Local homeomorphism

This article defines a property of continuous maps between topological spaces

Definition

Let $X$ and $Y$ be topological spaces. A surjective continuous map $f:X\to Y$ is termed a local homeomorphism if the following are true:

• It is an open map
• Every $x\in X$ has an open neighbourhood $U$ such that $f{|}_U$ is a homeomorphism to its image

Relation with other properties

Stronger properties

• Covering map
• Etale map (also sometimes called a sheaf map, though that term has other meanings)

Weaker properties

• Open map
• Quotient map