CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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 $T_0$ 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 in locale theory.
There are many ways to say that a space $X$ is compact. The first is perhaps the most common:
$X$ is compact iff for every collection of open subsets whose union is $X$ (i.e. which covers $X$), there is a (Kuratowski)-finite subcollection which also covers $X$ (i.e. a finite sub-cover).
If excluded middle is assumed, this is easily seen to be equivalent to:
$X$ is compact iff for any collection of closed subsets 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:
$X$ is compact iff every ultrafilter $\mathcal{U}$ (or ultranet $\nu$) on $X$ converges to some point $x \in X$, meaning that $\mathcal{U}$ 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 proper filter or equivalently (see at eventuality filter) of net convergence .
$X$ is compact iff every proper filter/net on $X$ has a convergent proper refinement/subnet.
This is equivalent to the characterization given in the Idea-section above:
$X$ is compact iff every proper filter $\mathcal{U}$ (or net $\nu$) on $X$ has a cluster point $x$, meaning that every element of $\mathcal{U}$ 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
$X$ is compact iff given any directed collection of opens whose union is $X$ (a directed open cover), $X$ belongs to the collection.
As the union is the coproduct in the category of open subsets $Op(X)$, we can also say
$X$ is compact iff it is a compact object in $Op(X)$.
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):
$X$ is compact iff for any space $Y$, the projection map $X \times Y \to Y$ 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). This equivalence is also true for locales, by way of proper maps; see below.
Closely related to the previous definition, a logical characterisation of compactness is used in Abstract Stone Duality:
$X$ is compact iff for any space $Y$ and any open subset $U$ of $X \times Y$, the subset
is open in $Y$.
To remove it from dependence on points, we can also write the definition like this:
$X$ is compact iff 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:
for every open $V$ in $Y$.
A dual condition is satisfied by an overt space.
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 $n$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).
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. 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 $T_1$, 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.
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.
The Heine-Borel theorem asserts that a subspace $S \subset \mathbb{R}^n$ of a Cartesian space is compact precisely if it is closed and bounded.
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.
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 $X$ is compact if and only if the unique map $X\to 1$ 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 $f_*$ 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.