nLab
local homeomorphism

Contents

Idea

A continuous map f:XYf : 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 XX as an étale space over YY.

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:EBp : E \to B between topological spaces (a morphism in Top) such that

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

or equivalently

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

See also etale space.

Examples

For YY any topological space and for SS any set regarded as a discrete space, the projection

X×SX X \times S \to X

is a local homeomorphism.

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

X:= iU i X := \coprod_i U_i

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

iU iY \coprod_i U_i \to Y

this is a local homeomorphism.

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

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