nLab
proper map

This entry is about the concept in topology. For variants see at proper morphism.

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Theorems

Contents

Definition

Definition

(proper maps)

A continuous function f:(X,τ X)(Y,τ Y)f \colon (X, \tau_X) \to (Y, \tau_Y) is called proper if for CYC \in Y a compact topological subspace of YY, then also its pre-image f 1(C)f^{-1}(C) is compact in XX.

Properties

Proposition

(maps from compact spaces to Hausdorff spaces are closed and proper)

Let f:(X,τ X)(Y,τ Y)f \colon (X, \tau_X) \longrightarrow (Y, \tau_Y) be a continuous function between topological spaces such that

  1. (X,τ X)(X,\tau_X) is a compact topological space;

  2. (Y,τ Y)(Y,\tau_Y) is a Hausdorff topological space.

Then ff is

  1. a closed map;

  2. a proper map (def. 1))

Definition

(locally compact topological space)

A topological space XX is called locally compact if for every point xXx \in X and every open neighbourhood U x{x}U_x \supset \{x\} there exists a smaller open neighbourhood V xU xV_x \subset U_x whose topological closure is compact (def. \ref{CompactTopologicalSpace}) and still contained in UU:

{x}V xCl(V x)compactU x. \{x\} \subset V_x \subset \underset{\text{compact}}{Cl(V_x)} \subset U_x \,.
Proposition

(proper maps to locally compact spaces are closed)

Let

  1. (X,τ X)(X,\tau_X) be a topological space,

  2. (Y,τ Y)(Y,\tau_Y) a locally compact topological space according to def. 2,

  3. f:XYf \colon X \to Y a continuous function.

Then:

If ff is a proper map (def. 1), then it is a closed map.

Created on May 12, 2017 16:30:05 by Urs Schreiber (46.183.103.8)