nLab open subspaces of compact Hausdorff spaces are locally compact

Contents

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

(open subspaces of compact Hausdorff spaces are locally compact)

Every open topological subspace XopenKX \underset{\text{open}}{\subset} K of a compact Hausdorff space KK is a locally compact topological space.

In particular every compact Hausdorff space itself is locally compact.

Proof

Let xXx \in X be a point and let U xXU_x \subset X an open neighbourhood. We need to produce a small open neighbourhood whose closure is compact and still contained in U xU_x.

By the nature of the subspace topology there exists an open subset V xKV_x\subset K such that U x=XV xU_x = X \cap V_x. Since XX is assumed to be open, it follows that UU is also open as a subset of KK. Since compact Hausdorff spaces are normal it follows (by this prop.) that there exists a smaller open neighbourhood W xKW_x \subset K whose topological closure is still contained in U xU_x, and since closed subspaces of compact spaces are compact:

{x}W xCl(W x)cptV xK. \{x\} \subset W_x \subset \underset{\text{cpt}}{Cl(W_x)} \subset V_x \subset K \,.

The intersection of this situation with XX is the required smaller compact neighbourhood Cl(W x)XCl(W_x) \cap X:

{x}W xXCl(W x)cptXU xX. \{x\} \subset W_x \cap X \subset \underset{\text{cpt}}{Cl(W_x)} \cap X \subset U_x \subset X \,.
Remark

Conversely, every locally compact Haudorff space XX arises as in prop. , since it may be considered an open subspace in its one-point compactification X{}X \sqcup \{\infty\}. See there this example.

Last revised on June 3, 2017 at 08:56:29. See the history of this page for a list of all contributions to it.