nLab local homeomorphism

Context

Topology

topology

algebraic topology

Contents

Idea

A continuous map $f : X \to Y$ between topological spaces is called a local homeomorphism if restricted to a neighbourhood of every point in its domain it becomes a homeomorphism.

One also says that this exhibits $X$ as an étale space over $Y$.

Notice that, despite the similarity of terms, local homeomorphisms are, in general, not local isomorphisms in any natural way. See the examples below.

Definition

A local homeomorphism is a continuous map $p : E \to B$ between topological spaces (a morphism in Top) such that

• for every $e \in E$, there is an open set $U \ni e$ such that the image $p_*(U)$ is open in $B$ and the restriction of $p$ to $U$ is a homeomorphism $p|_U: U \to p_*(U)$,

or equivalently

• for every $e \in E$, there is a neighbourhood $U$ of $e$ such that the image $p_*(U)$ is a neighbourhood of $p(e)$ and $p|_U: U \to p_*(U)$ is a homeomorphism.

Examples

For $Y$ any topological space and for $S$ any set regarded as a discrete space, the projection

$X \times S \to X$

is a local homeomorphism.

For $\{U_i \to Y\}$ an open cover, let

$X := \coprod_i U_i$

be the disjoint union space of all the pathches. Equipped with the canonical projection

$\coprod_i U_i \to Y$

this is a local homeomorphism.

In general, for every sheaf $A$ of sets on $Y$; there is a local homeomorphism $X \to Y$ such that over any open $U \hookrightarrow X$ the set $A(U)$ is naturally identified with the set of sections of $Y \to X$. See étale space for more on this.

Revised on March 19, 2012 11:11:10 by Urs Schreiber (89.204.155.155)