nLab
compactum

Idea

A compactum, or compact Hausdorff space, is a space in which every limit that should exist does exist and does so uniquely.

One can define this literally for topological spaces, or in terms of convergence, or for locales; although these are all different contexts, the resulting notion of compactum is (at least assuming the axiom of choice) always the same. Interestingly, there is even an algebraic definition, not one that uses only finitary operations, but one which uses a monad.

Definitions

If you know what a compact space is and what a Hausdorff space is, then you know what a compact Hausdorff space is, so let's be fancy.

Given a set $S$, let $\beta S$ be the set of ultrafilters on $S$. Note that $\beta$ is an endofunctor on Set; every function $f:S\to T$ induces a function $\beta f:\beta S\to \beta T$ using the usual application of functions to filters. In fact, $\beta$ is a monad; it comes with a natural (in $S$) unit ${\eta }_S:S\to \beta S$, which maps a point $x$ to the principal ultrafilter that $x$ generates, and multiplication ${\mu }_S:\beta \beta S\to \beta S$, which maps an ultrafilter $U$ on ultrafilters to the ultrafilter of sets whose principal ultrafilters of ultrafilters belong to $U$. That is,

• $A\in \eta x\iff x\in A$, so $\eta x=\{A\subseteq S\mid x\in A\}$;
• $A\in \mu U\iff \{F\in \beta S\mid A\in F\}\in U$.

Then a compactum is simply an algebra for this monad; that is, a set $X$ together with a function $\lim :\beta X\to X$, such that

• each point $x$ is the limit ($\lim$) of the principal ultrafilter $\eta x$, and
• given an ultrafilter $U$ on ultrafilters, the limit of $\mu U$ is the limit of $(\beta \lim )U$.

It is then a theorem that this $\lim$ generates a convergence on $S$ that is compact, Hausdorff, and topological. The converse, that every compact Hausdorff topological convergence is of this form, is equivalent to the ultrafilter principle.

Every compact Hausdorff space is regular and sober and so defines a compact regular locale. Again, the axiom of choice gives us a converse: every compact regular locale is spatial and so comes from a compactum.

Probably this is also equivalent to the ultrafilter principle, but I need to check.

Note that every compact Hausdorff space (topological or localic) is not only regular but also normal.

Compacta as algebras

Under construction…

The category of compact Hausdorff spaces is monadic over $\mathrm{Set}$. In other words, there is a monad $\beta :\mathrm{Set}\to \mathrm{Set}$ and an equivalence $\mathrm{CH}\to \mathrm{Alg}(\beta )$ which respects the underlying functors to $\mathrm{Set}$ up to isomorphism:

$$(\mathrm{CH}\stackrel{U}{\to }\mathrm{Set})\cong (\mathrm{CH}\to \mathrm{Alg}(\beta )\stackrel{U}{\to })$$(CH \stackrel{U}{\to} Set) \cong (CH \to Alg(\beta) \stackrel{U}{\to})

The functor $\beta$ takes a set $S$ to the set of ultrafilters on $S$, which may be identified with Boolean algebra homomorphisms $\varphi :2^S\to 2$. Alternatively, each ultrafilter may be identified with a proper subset $𝒰={\varphi }^{-1}(1)\subseteq P(S)$ of the power set of $S$ which is

• Upward-closed: $A\in 𝒰$ and $A\subseteq B\subseteq S$ implies $B\in 𝒰$;

• Closed under finite intersections: ($A\in 𝒰$ and $B\in 𝒰)\mathrm{implies}$A \cap B \in \mathcal{U}$.

This construction is functorial: a function $f:S\to T$ evidently induces a function $\beta (f):\beta (S)\to \beta (T)$:

$$\mathrm{Bool}(2^f,2):\mathrm{Bool}(2^S,2)\to \mathrm{Bool}(2^T,2)$$Bool(2^f, 2): Bool(2^S, 2) \to Bool(2^T, 2)

The unit of the monad $\beta$ takes each $s\in S$ to the Boolean algebra homomorphism ${\mathrm{ev}}_s:2^S\to 2$. Let ${\eta }_S$ denote this component of the unit.

Observe also that for any Boolean algebra $B$, there is a canonical Boolean algebra homomorphism

$${\epsilon }_B:B\to 2^{\mathrm{Bool}(B,2)}:b\mapsto {\mathrm{ev}}_b$$\varepsilon_B: B \to 2^{Bool(B, 2)}: b \mapsto ev_b

The multiplication of the monad $\beta$ is the map

$$\mathrm{Bool}({\epsilon }_{2^S},2):\mathrm{Bool}(2^{\mathrm{Bool}(2^S,2)},2)\to \mathrm{Bool}(2^S,2)$$Bool(\varepsilon_{2^S}, 2): Bool(2^{Bool(2^S, 2)}, 2) \to Bool(2^S, 2)

Next, if $X$ is compact Hausdorff, $UX$ its underlying set, there is an algebra structure

$$\xi :\beta (UX)\to UX$$\xi: \beta(U X) \to U X

which maps each ultrafilter on $UX$ to the unique point it converges to in $X$.

(To be continued…)

In weak foundations

In the absence of the axiom of choice, and especially in constructive mathematics, the best definition of compactum seems to be a compact regular locale. That is, it is the category of compact regular locales that has all of the nice properties, forming a nice category of spaces, and that has the desired examples, such as the unit interval. (See the discussion at Tychonoff theorem for an example of how the category of compact Hausdorff topological spaces might fail to be nice; see Frank Waaldijk’s PhD thesis (pdf) for a thorough discussion of what is needed to make the unit interval a compact Hausdorff topological space.)

The monadic definition, in particular, falls quite flat without some form of the axiom of choice; even excluded middle and COSHEP are powerless here. In fact, it is quite consistent to assume that every ultrafilter is principal (a strong denial of the ultrafilter principle), in which case $\beta$ is the identity monad. Then a compactum would be just a set if that were the definition used.

On the other hand, it is the monadic definition that gives an algebraic category with a nice relationship to Set. Without the ultrafilter principle, there is no reason to think that the set-of-points functor from compact regular locales to sets is even continuous.

Stone–Čech compactification

The monad $\beta$ is commutative, so every $\beta S$ is itself a compactum. The functor $\beta :\mathrm{Set}\to \mathrm{Comp}$ is left adjoint to the forgetful functor $\mathrm{Comp}\to \mathrm{Set}$. Assuming the ultrafilter principle, this can be generalised to a functor $\beta :\mathrm{Top}\to \mathrm{Comp}$ (identifying a set with its discrete space) that is left adjoint to the forgetful functor $\mathrm{Comp}\to \mathrm{Top}$. This (or at least its restriction to Tychonoff spaces) is the Stone–Čech compactification functor. We have a similar Stone-Čech compactification functor $\mathrm{Loc}\to \mathrm{Comp}$; we do not need the ultrafilter principle here if $\mathrm{Comp}$ is defined in terms of locales.

Related concepts

• vanishing at infinity