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proper maps to locally compact spaces are closed

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

Background

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 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 \,.

Statement and proof

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. 1,

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

Then:

If ff is a proper map, then it is a closed map.

Proof

Let CXC \subset X be a closed subset. We need to show that every yY\f(C)y \in Y \backslash f(C) has an open neighbourhood U y{y}U_y \supset \{y\} not intersecting f(C)f(C) (by this prop.).

By local compactness of (Y,τ Y)(Y,\tau_Y) (def. 1), yy has an open neighbourhood V yV_y whose topological closure Cl(V y)Cl(V_y) is compact. Hence since ff is proper, also f 1(Cl(V y))Xf^{-1}(Cl(V_y)) \subset X is compact. Then also the intersection Cf 1(Cl(V y))C \cap f^{-1}(Cl(V_y)) is compact, and hence so is

f(Cf 1(Cl(V y)))=f(C)(Cl(V))Y. f(C \cap f^{-1}(Cl(V_y))) = f(C) \cap (Cl(V)) \; \subset Y \,.

This is also a closed subset, since compact subspaces of Hausdorff spaces are closed. Therefore

U yV y\(f(C)(Cl(V y)))=V y\f(C) U_y \coloneqq V_y \backslash ( f(C) \cap (Cl(V_y)) ) = V_y \backslash f(C)

is an open neighbourhod of yy not intersecting f(C)f(C).

Revised on May 16, 2017 13:49:24 by Urs Schreiber (131.220.184.222)