A compactum, or compact Hausdorff space, is a space in which every limit that should exist does exist and does so uniquely. Slightly more generally, one sometimes wants to speak of a compact regular space.
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.
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. (Full justifications will be provided in section on compacta as algebras.)
Given a set , let be the set of ultrafilters on . Note that is an endofunctor on Set; every function induces a function using the usual application of functions to filters. In fact, is a monad; it comes with a natural (in ) unit , which maps a point to the principal ultrafilter that generates, and multiplication , which maps an ultrafilter on ultrafilters to the ultrafilter of sets whose principal ultrafilters of ultrafilters belong to . That is,
Then a compactum is simply an algebra for this monad; that is, a set together with a function , such that
It is then a theorem that this generates a convergence on 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.
Throughout this section, will be used to denote the category of compact Hausdorff spaces (compacta).
for a subset of . Notice is empty. Indeed, defines a Boolean algebra map
so that in particular , which immediately implies that the form a basis of a topology.
In fact, is the component at of the counit of the adjunction , as a morphism in . If is a function, the commutativity of the naturality square
implies that if is a basic (cl)open, then so is . It then follows that is continuous.
These results show that the monad lifts through the forgetful functor .
The unit of the monad is given componentwise by functions
where takes to the principal ultrafilter
It is evident that is injective.
The injection exhibits as a dense subset of .
If is a basic open neighborhood containing an ultrafilter , then is nonempty and hence contains some , which is to say or that .
Let be distinct ultrafilters, so there is with and . Then and are disjoint neighborhoods which contain and respectively.
It is enough to show that if is a collection of opens such that the union of any finite subcollection is a proper subset, then the union of is also proper.
If covers , it admits a refinement by basic clopens also covering , and thus we may assume WLOG that consists of basic clopens . If every finite union of elements of is a proper subset of , then every finite intersection is nonempty, so that the generate a filter, which is contained in some ultrafilter . This lies outside the union of all the ‘s.
Let be a topological space, and an ultrafilter on the underlying set . We say converges to a point (in symbols, ) if the neighborhood filter of is contained in .
Convergence defines a relation from to .
If is Hausdorff, then the relation is well-defined, or functional (i.e., there is at most one point to which a given ultrafilter converges).
If , then there are disjoint neighborhoods , of and . We cannot have both and (otherwise would be an element of ), so at most one of the neighborhood filters can be contained in .
If is compact, then the relation is total (i.e., there exists a point to which a given ultrafilter converges).
If not, then for each there is an open neighborhood that does not belong to . Then . Some finite number of neighborhoods covers . Then , which is a contradiction.
If is compact Hausdorff, then the function is continuous.
Let be an open neighborhood of ; we must show that contains an open neighborhood of any of its points (i.e., ultrafilters such that ). Since is (Hausdorff regular), we may choose a neighborhood whose closure is contained in . Then is an open neighborhood of in , and we claim .
For this, we must check that if and , then . But if , then , whence implies . This contradicts , i.e., contradicts , since .
If is a set and is a compact Hausdorff space, then any function can be extended (along ) to a continuous function .
We define to be the composite
The square commutes by naturality of , and commutativity of the triangle simply says that the ultrafilter converges to , or that , which reduces to the tautology that for every neighborhood .
Proposition 7 shows that for any function to a compact Hausdorff space, there exists continuous such that . All that remains is to establish uniqueness of such . But if two maps to a Hausdorff space agree on a dense subspace, in this case the subspace by Proposition 1, then they must be equal. Indeed, the pullback of the closed diagonal defines a closed subspace of ,
and contains a dense subspace , therefore ; i.e., the equalizer of and is all of , hence these two maps are equal.
We recall hypotheses of Beck’s precise monadicity theorem: a functor is monadic if and only if
has a left adjoint,
reflects isomorphisms: a morphism of is an isomorphism if is an isomorphism in ,
has, and preserves, coequalizers of parallel pairs that are -split. (We say
is -split if there is a coequalizer
that is split in : there exists and such that , , and .)
In the case where , we have the following useful lemma:
Suppose given a coequalizer in
split by , (so that , , ). Let be the image of , and let be the equivalence relation given by the kernel pair of . Then , the relational composite given by taking the image of the span composite .
Clearly since , and we have and by symmetry and transitivity of . In the other direction, suppose , so that . Put and (so that and ), and put . Clearly then . Moreover,
so that , or . Hence , and we have shown .
The forgetful functor is monadic.
By theorem 1, has a left adjoint. Since bijective continuous maps between compact Hausdorff spaces are homeomorphisms, we have that reflects isomorphisms. Finally, suppose is a -split pair of morphisms in ; let be their coequalizer in , given by a suitable quotient space. Being a quotient of a compact space, is compact. Since is a full subcategory of , the map is a coequalizer in once we prove the following claim:
Furthermore, since the forgetful functor has a right adjoint (given by taking indiscrete topologies on sets), the underlying function of (again denoted ) is the coequalizer of in , so that would preserve the claimed coequalizer.
In other words, to complete the proof, it suffices to verify the claim. Letting be the kernel pair of in , to show is Hausdorff, it suffices to prove that the equivalence relation in is closed. Let be the image of . By lemma 1, the subset of coincides with the subset . Now is the image of the compact space under the continuous map , so is a closed subset of . Similarly is a closed subset of . Under their subspace topologies, their fiber product is compact, and so its image under the (continuous) span composite is also closed. This completes the proof.
Every Hausdorff space, hence every compactum, satisfies the separation axiom . As is usual with separation axioms, we can also look for a non- version. A priori, this is a compact preregular space; however, since every such space is regular, we can speak instead of a compact regular space.
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 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.
By general nonsense, every , regarded as a free -algebra, is a compactum, and the functor
is left adjoint to the forgetful functor . Assuming the ultrafilter principle, this functor extends to a functor (identifying a set with its discrete space) that is left adjoint to the forgetful functor . This is the Stone–Čech compactification functor (N.B.: for many authors, Stone–Čech compactification refers to the restriction of this functor to Tychonoff spaces , which are precisely those spaces where the unit is an embedding so that we have a compactification in the technical sense).
The monad is given on objects by ; this lands in compact Hausdorff spaces. Let be the closure of the image of the unit ; under the ultrafilter principle, is compact Hausdorff.
If is a Tychonoff space, then the unit is a subspace embedding, so that is a cogenerator in the category of Tychonoff spaces. (In particular, is an embedding if is compact Hausdorff, so is also a cogenerator in the category of compact Hausdorff spaces.)
The natural map is universal among maps from to compact Hausdorff spaces, thus giving a left adjoint to the (fully faithful) forgetful functor .
Let be a map, where is a compact Hausdorff space. Since is a cogenerator in the category of compact Hausdorff spaces, the unit for the codensity monad ,
is a continuous injection (and hence a closed subspace embedding, since is compact Hausdorff). Let , and consider the pullback square
From an evident naturality square for the unit , we have a map , i.e., the map factors through the closed subspace . Therefore factors as
and since , we conclude that factors through . And moreover, there is at most one such that , because maps onto a dense subspace of , and dense subspaces are epic in the category of Hausdorff spaces. This completes the proof.
We have a similar Stone–Čech compactification functor ; we do not need the ultrafilter principle here if is defined in terms of locales.
The category of compact Hausdorff spaces and continuous maps is
Barr exact, since is monadic,
an extensive category, and
a total category (by monadicity over ), and
a cototal category (because it is complete, well-powered, and has a cogenerator given by the unit interval ).
From the first two properties, it follows that is a pretopos, meaning that enjoys the same finitary exactness properties that hold in a topos; in particular, first-order intuitionistic logic may be enacted within .