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.
One often wishes to study compact Hausdorff spaces. For locales, one usually speaks of compact regular locales; these are equivalent (since every locale is and hence if regular, while every Hausdorff space is if compact) since regularity is easier to formulate and handle than Hausdorffness in locale theory.
There are many ways to say that a space is compact. The first is perhaps the most common:
If excluded middle is assumed, this is easily seen to be equivalent to:
This is equivalent to the characterization given in the Idea-section above:
is compact iff every proper filter (or net ) on has a cluster point , meaning that every element of meets (has inhabited intersection with) every neighbourhood of (or is frequently in every neighbourhood of ).
is compact iff given any directed collection of opens whose union is (a directed open cover), belongs to the collection.
is compact iff it is a compact object in .
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). This equivalence is also true for locales, by way of proper maps; see below.
is compact iff for any space and any open subset of , the subset
is open in .
To remove it from dependence on points, we can also write the definition like this:
for every open in .
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 Lab 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).
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 , 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.
In synthetic topology, where ‘space’ means simply ‘set’ (or type, i.e. the basic objects of our foundational system), one natural notion of “compact space” is a covert set, i.e. a set whose discrete topology is covert. This includes the expected examples in various gros toposes.
A space is compact if and only if the unique map is proper. Thus, properness is a “relativized” version of compactness.
For topological spaces, this is either a definition of “proper map” (closed with compact fibers) or follows from the above characterization of compactness in terms of projections being closed maps (if proper maps are defined to be those that are universally closed). For locales, it follows from the definition of proper map (a closed map such that preserves directed joins) and the fact that compact locales are automatically covert (see covert space for a proof).
For proper base change theorem e.g.