Local homeomorphism

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 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 and $f(U)$ is itself an open subset of $Y$.

Some variants of the definition of local homeomorphism also require the map to be surjective.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
covering map				|FULL LIST, MORE INFO
etale map	(also sometimes called a sheaf map, though that term has other meanings)			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
open map	image of any open subset of the domain is open	local homeomorphism implies open map		|FULL LIST, MORE INFO
quotient map (if surjective)