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 ${p}_{*}\left(U\right)$ is open in $B$ and the restriction of $p$ to $U$ is a homeomorphism $p{\mid }_{U}:U\to {p}_{*}\left(U\right)$,

or equivalently

• for every $e\in E$, there is a neighbourhood $U$ of $e$ such that the image ${p}_{*}\left(U\right)$ is a neighbourhood of $p\left(e\right)$ and $p{\mid }_{U}:U\to {p}_{*}\left(U\right)$ is a homeomorphism.

Examples

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

$X×S\to X$X \times S \to X

is a local homeomorphism.

For $\left\{{U}_{i}\to Y\right\}$ an open cover, let

$X:=\coprod _{i}{U}_{i}$X := \coprod_i U_i

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

$\coprod _{i}{U}_{i}\to Y$\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 $U↪X$ the set $A\left(U\right)$ 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)