nLab
locally proper map

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

The relative version of a locally compact topological space is a continuous map that behaves like a ‘family of locally compact spaces’.

Definition

Definition

A continuous function f:XYf \colon X \to Y is called locally proper if for any xXx \in X and neighbourhood VxV \ni x there is a neighbourhood AxA\ni x with AVA \subset V and a neighbourhood Uf(x)U \ni f(x) such that AUA \to U is a proper map.

Examples

  • The inclusion map of a locally closed subset CXC\hookrightarrow X in a topological space XX is locally proper.

  • A topological space XX is a locally compact space if and only if X*X\to \ast is locally proper.

Properties

Proposition

Let f:XYf\colon X\to Y and g:YZg\colon Y\to Z be locally proper. Then gfg\circ f is locally proper. If WYW\to Y is any continuous function then W× YZWW\times_Y Z \to W is locally proper. (ie locally proper maps are closed under composition and stable under pullback. Hence they form a singleton coverage)

As a corollary, one has that for a locally compact space XX, every locally closed subspace of XX is locally compact.

Proposition

Let f:XYf\colon X\to Y be a continuous function. Then if

  • there is an open covering 𝒰={U i}\mathcal{U} = \{U_i\} of XX such that f| U i:U iYf\big|_{U_i} \colon U_i \to Y is locally proper for all ii,

or

  • there is an open covering 𝒱={V j}\mathcal{V} = \{V_j\} of YY such that f 1(V j)V jf^{-1}(V_j) \to V_j is locally proper for all jj,

then ff is locally proper. The converse implications also hold

Recall that a continuous map g:YZg\colon Y \to Z of topological spaces is called separated if YY× ZYY\to Y\times_Z Y is a closed embedding.

Proposition

If f:XYf\colon X\to Y and g:YZg\colon Y\to Z are continuous maps, gg is separated and gfg\circ f is locally proper, then ff is locally proper

Corollary

Every continuous map from a locally compact Hausdorff space to a Hausdorff space is separated and locally proper.

The following proposition generalises the well-known result that compact Hausdorff spaces are locally compact.

Proposition

Every separated and proper map is locally proper

The proper base change theorem also holds for locally proper maps if one uses direct image with compact support instead of ordinary direct image

References

  • Olaf M. Schnürer and Wolfgang Soergel, Proper base change for separated locally proper maps, Rend. Sem. Mat. Univ. Padova 135 (2016) pp 223-250. doi:10.4171/RSMUP/135-13, arXiv:1404.7630

Last revised on January 19, 2018 at 08:25:50. See the history of this page for a list of all contributions to it.