nLab
local homeomorphism

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 $p{\mid }_U:U\to p_{*}(U)$,

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 $p{\mid }_U:U\to p_{*}(U)$ is a homeomorphism.

See also etale space.

Examples

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

$$X\times S\to X$$X \times S \to X

is a local homeomorphism.

For $\{U_i\to Y\}$ 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 $U\hookrightarrow X$ the set $A(U)$ is naturally identified with the set of sections of $Y\to X$. See étale space for more on this.

Related concepts

• étale map, étale morphism

• locally homeomorphic geometric morphism