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maps from compact spaces to Hausdorff spaces are closed and proper

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

(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.

Proof

For the first statement, we need to show that if CXC \subset X is a closed subset of XX, then also f(C)Yf(C) \subset Y is a closed subset of YY.

Now

  1. since closed subsets of compact spaces are compact it follows that CCC \subset C is also compact;

  2. since continuous images of compact spaces are compact it then follows that f(C)Yf(C) \subset Y is compact;

  3. since compact subspaces of Hausdorff spaces are closed it finally follow that f(C)f(C) is also closed in YY.

For the second statement we need to show that if CYC \subset Y is a compact subset, then also its pre-image f 1(C)f^{-1}(C) is compact.

Now

  1. since compact subspaces of Hausdorff spaces are closed it follows that CsubseYC \subse Y is closed;

  2. since pre-images under continuous of closed subsets are closed, also f 1(C)Xf^{-1}(C) \subset X is closed;

  3. since closed subsets of compact spaces are compact, it follows that f 1(C)f^{-1}(C) is compact.

Consequences

Corollary

(continuous bijections from compact spaces to Hausdorff spaces are homeomorphisms)

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.

  3. f:XYf \;\colon\; X \longrightarrow Y is a bijection of sets.

Then ff is a homeomorphism, i. e. its inverse function YXY \to X is also a continuous function.

In particular then both (X,τ X)(X,\tau_X) and (Y,τ Y)(Y, \tau_Y) are compact Hausdorff spaces.

Proof

Write g:YXg \colon Y \to X for the inverse function of ff.

We need to show that gg is continuous, hence that for UXU \subset X an open subset, then also its pre-image g 1(U)Yg^{-1}(U) \subset Y is open in YY. By passage to complements, this is equivalent to the statement that for CXC \subset X a closed subset then the pre-image g 1(C)Yg^{-1}(C) \subset Y is also closed in YY.

But since gg is the inverse function to ff, its pre-images are the images of ff. Hence the last statement above equivalently says that ff sends closed subsets to closed subsets. This is true by prop. 1.

Remark

The idea captured by corollary 1 is that Hausdorffness is about having “enough” open sets whilst compactness is about having “not too many”. Thus a compact Hausdorff space has both “enough” and “not too many”. This theorem says that both conditions are at their limit: if we try to have more open sets, we lose compactness. If we try to have fewer open sets, we lose Hausdorffness.

Revised on June 21, 2017 04:49:51 by Urs Schreiber (131.220.184.222)