nLab
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces

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

Statement

Proposition

Let (X,τ)(X,\tau) be a compact Hausdorff topological space and let YXY \subset X be a topological subspace. Then the following are equivalent:

  1. YXY \subset X is a closed subspace;

  2. YY is a compact topological space.

Proof

The two directions to be proven are

  1. closed subsets of compact spaces are compact

  2. compact subspaces of Hausdorff spaces are closed.

See the proofs there.

Revised on May 15, 2017 12:14:12 by Urs Schreiber (195.37.209.183)