nLab
compact space

Idea

A topological space (or more generally 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.

Definitions

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

Definition

For every collection of open sets whose union is $X$ (which covers $X$), there is a (Kuratowski)-finite subcollection which also covers $X$.

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

Definition

For any collection of closed sets of $X$ 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

Every ultrafilter $𝒰$ (or ultranet $\nu$) on $X$ converges to some point $x\in X$, meaning that $𝒰$ contains the filter of neighborhoods of $x$ (or that $\nu$ is eventually in any neighbourhood of $x$).

In any case, compactness can be characterized in terms of filter (or net) convergence:

Definition

Every proper filter (or net) on $X$ has a convergent proper refinement (or subnet).

This is equivalent to the characterization given in Idea above:

Definition

Every proper filter $𝒰$ (or net $\nu$) on $X$ has a cluster point $x$, meaning that every element of $𝒰$ meets (has inhabited intersection with) every neighbourhood of $x$ (or $\nu$ is frequently in every neighbourhood of $x$).

While the usual definitions (1&2) are for topological spaces, the convergence definitions (3–5) 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

Given any directed collection of opens whose union is $X$, $X$ belongs to the collection.

As the union is the coproduct in the category of open subsets $\mathrm{Op}(X)$, we can also say

Definition

The ambient space $X$ is a compact object in $\mathrm{Op}(X)$.

At least assuming the axiom of choice, compactness can also be characterized in terms of universal closure:

Definition

For any space $Y$, the projection map $X\times Y\to Y$ is closed.

I’m not entirely sure that the axiom of choice is needed to prove the last equivalence. It’s easy (and classical) that the projection map is closed if $X$ is compact. For the other direction, I know a proof which involves taking $Y$ to be the space of ultrafilters on the underlying set of $Y$, but there may be some other choice(s) for $Y$ which allows one to get around AC.

A logical characterisation of compactness is used in Abstract Stone Duality:

Definition

For any space $Y$ and any open subset $U$ of $X\times Y$, the subset

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

is open in $Y$.

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

Definition

Given any space $Y$ and any open $U$ in $X\times Y$, there exists an open ${\forall }_{X}U$ in $Y$ that satisfies the universal property of universal quantification?:

$$V\subseteq {\forall }_{X}U\iff X\times V\subseteq U$$V \subseteq \forall_X U \;\Leftrightarrow\; X \times V \subseteq U

for every open $V$ in $Y$.

A dual condition is satisfied by an overt space.

Standard Results

• 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.

• The direct image of a compact subspace under a continuous map is compact. Thus any topological space becomes a bornological space? by taking the bounded sets to be compact subspaces.

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

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

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 nLab 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).