nLab
compact space

Compact spaces

Idea

A topological space (or more generally: a convergence space) is compact if everything converges as much as possible. It is a kind of ultimate topological expression of the general idea of a space being “closed and bounded”: every net must accumulate somewhere in the space; by boundedness it cannot escape, and by closure the point is in the space. There is also a notion of compactness for locales.

It is also common to work with compact subsets of a space. These are those subsets which are compact spaces with the subspace topology.

One often wishes to study compact Hausdorff spaces. For locales, one usually speaks of compact regular locales; these are equivalent (since every locale is T 0T_0 and hence T 3T_3 if regular, while every Hausdorff space is T 3T_3 if compact) since regularity is easier to formulate and handle than Hausdorffness in locale theory.

Definitions

There are many ways to say that a space XX is compact. The first is perhaps the most common:

Definition

XX is compact if for every collection of open subsets whose union is XX (i.e. which covers XX), there is a (Kuratowski)-finite subcollection which also covers XX (i.e. a finite sub-cover).

If excluded middle is assumed, this is easily seen to be equivalent to:

Definition

XX is compact if for any collection of closed subsets of XX whose intersection is empty, some finite subcollection also has empty intersection.

If the ultrafilter theorem (a weak form of the axiom of choice) is assumed, compactness can be characterized in terms of ultrafilter (or ultranet) convergence:

Definition

XX is compact if every ultrafilter 𝒰\mathcal{U} (or ultranet ν\nu) on XX converges to some point xXx \in X, meaning that 𝒰\mathcal{U} contains the filter of neighborhoods of xx (or that ν\nu is eventually in any neighbourhood of xx).

In any case, compactness can be characterized in terms of proper filter or equivalently (see at eventuality filter) of net convergence .

Definition

XX is compact if every proper filter/net on XX has a convergent proper refinement/subnet.

This is equivalent to the characterization given in the Idea-section above:

Definition

XX is compact if every proper filter 𝒰\mathcal{U} (or net ν\nu) on XX has a cluster point xx, meaning that every element of 𝒰\mathcal{U} meets (has inhabited intersection with) every neighbourhood of xx (or ν\nu is frequently in every neighbourhood of xx).

While the usual definitions (1&2) are for topological spaces, the convergence definitions (35) make sense in any convergence space.

The definition (1) also works for locales, since it refers only to the frame of open sets. An equivalent way to phrase it is

Definition

XX is compact if given any directed collection of opens whose union is XX (a directed open cover), XX belongs to the collection.

As the union is the coproduct in the category of open subsets Op(X)Op(X), we can also say

Definition

XX is compact it it is a compact object in Op(X)Op(X).

Compactness is equivalent to the condition of being “stably closed” (and it is this condition which suggests the correct notion of proper map in algebraic geometry and elsewhere):

Definition

XX is compact if for any space YY, the projection map X×YYX \times Y \to Y out of their Cartesian product is closed (see e.g. Milne, section 17).

Contrary to possible appearance, the equivalence of this with definition 1 does not require the axiom of choice; see this MO question and answers, as well as this page. See also the page compactness and stable closure (under construction).

Closely related to the previous definition, a logical characterisation of compactness is used in Abstract Stone Duality:

Definition

XX is compact if for any space YY and any open subset UU of X×YX \times Y, the subset

XU={b:Y|a:X,(a,b)U} \forall_X U = \{ b : Y \;|\; \forall\; a: X,\; (a, b) \in U \}

is open in YY.

To remove it from dependence on points, we can also write the definition like this:

Definition

XX is compact if given any space YY and any open UU in X×YX \times Y, there exists an open XU\forall_X U in YY that satisfies the universal property of universal quantification:

V XUX×VU V \subseteq \forall_X U \;\Leftrightarrow\; X \times V \subseteq U

for every open VV in YY.

A dual condition is satisfied by an overt space.

Differing terminology

Some authors use “compact” to mean “compact Hausdorff” (a much nicer sort of space, and forming a much nicer category of spaces), and use the word “quasicompact” to refer to just “compact” as we are using it here. This custom seems to be prevalent among algebraic geometers, for example, and particularly so within Francophone schools.

But it is far from clear to me (Todd Trimble) that “quasicompact” is very well-established outside such circles (despite some arguments in favor of it), and using simply “compact” for the nicer concept therefore carries some risk of creating misunderstanding among mathematicians at large. My own habit at any rate is to say “compact Hausdorff” for the nicer concept, and I will continue using this on the nnLab until consensus is reached (if that happens).

Another term in usage is ‘compactum’ to mean a compact Hausdorff space (even when ‘compact’ is not used to imply Hausdorffness).

Properties

Proposition

Assuming the axiom of choice, the category of compact spaces admits all small limits. In any case, the category of compact locales admits all small limits. See also Tychonoff theorem.

Proposition

The direct image of a compact subspace under a continuous map is compact. A topological space becomes a bornological set by taking the bounded sets to be subsets contained in some compact subspace, and under this bornology, every continuous function is a bounded map.

If the spaces in question are T 1T_1, then the sets with compact closure also constitute a bornology and continuous maps become bounded. In a non-Hausdorff situation these bornologies might differ because the closure of a compact set need not be compact.

Proposition

A compact Hausdorff space must be normal. That is, the separation axioms T 2T_2 through T 4T_4 (when interpreted as an increasing sequence) are equivalent in the presence of compactness.

Proposition

The Heine-Borel theorem asserts that a subspace S nS \subset \mathbb{R}^n of a Cartesian space is compact precisely if it is closed and bounded.

Proposition

A discrete space is compact iff its underlying set is finite. In constructive mathematics, a discrete space is compact iff its underlying set is Kuratowski-finite.

References

For proper base change theorem e.g.

Revised on April 25, 2016 17:25:29 by Urs Schreiber (79.204.25.229)