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maps from compact spaces to Hausdorff spaces are closed and proper

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

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 CXC \subset X 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 CYC \subset 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. .

Remark

The idea captured by corollary 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.

Applications

In Cohomotopy and Cobordism theory

Pontryagin's theorem establishes a bijection

π n(X d)=π 0Maps(X d,S n)Cob Fr n(X d) \pi^n \big( X^d \big) \,=\, \pi_0 Maps \big( X^d, S^n \big) \overset {\simeq}{ \;\; \longrightarrow \;\; } Cob^n_{Fr} \big( X^d \big)

between

  1. the nn-Cohomotopy set of a closed (hence: compact) smooth manifold X dX^d;

  2. the cobordism classes of its normally framed submanifolds of codimension nn;

by taking the homotopy class of any map [XcS n][X \overset{c}{\to}S^n] into the n-sphere to the preimage of any regular point (say {0}S n\{0\} \subset S^n, for definiteness) of a smooth representative cc:

Σ cc 1({0}). \Sigma_c \;\coloneqq\; c^{-1}\big( \{0\} \big) \,.

To show that this construction is indeed injective, one needs that, similarly, the pre-image of any regular point of a smooth homotopy

X d×[0,1]ηS d X^d \times [0,1] \overset{\eta}{\longrightarrow} S^d

between two such smooth representatives c 0,c 1c_0, c_1 is a cobordism between Σ c 0\Sigma_{c_0} and Σ c 1\Sigma_{c_1}. But for a subspace of X d×[0,1]X^d \times [0,1] to constitute a cobordism between submanifolds of X dX^d it is necessary that its projection onto the [0,1][0,1]-factor is compact.

That this is implied here is guaranteed by the assumption that X dX^d is closed, hence compact, so that also the product space X d×[0,1]X^d \times [0,1] is compact:

With this and since the n-sphere is, of course, Hausdorff, Prop. implies that the maps c ic_i and η\eta above are all proper, hence that the corresponding pre-images (of singleton, hence compact, subspaces) are indeed compact, so that in particular the projection of the pre-image of η\eta to [0,1][0,1] is compact (since continuous images of compact spaces are compact).

Last revised on February 3, 2021 at 18:28:40. See the history of this page for a list of all contributions to it.