A pointfree description of convex compact subsets
In my thesis I needed to represent compact convex subsets of compact partially-ordered spaces without actually referring to points of the spaces. The motivation behind this is that the points of the Vietoris hyperspace (of compact partially-ordered spaces) are precisely the compact convex subsets and if we want to describe what the points are constructively, we need to be able to axiomatise such subsets without referring to the points of the original space.
Compact partially ordered spaces , i.e. compact spaces with the order closed in the product , can be equivalently described as compact regular bitopological spaces and that is the description I used in my thesis (the definition of regularity and compactness is given below). The two topologies and are just the topologies of open upsets and open downsets, respectively.
The key observation is that every compact convex subset of can be described in terms of its complement. Take the sets Since and are closed subsets we see that and . Conversely, for any pair , the complement of their union is compact and convex. However, there might be many pairs representing the same .
Finding a canonical representation of compact convex subsets amounts to trimming down the pairs of opens to the maximal such pairs, that is, whenever is equal to , for some , then it must be that and .
On the other hand, if we are given a pair of opens, we can easily find the canonical representation of the convex compact set they represent. Just take, the pair mentioned earlier.
So far we didn’t do anything special. So long we stick to bitopological spaces we can comfortably represent convex compact subsets as described above. The problem comes when we move to the pointfree setting. Instead of a space consisting of two topologies we have a pair of frames and (representing the two topologies) and two relations and . The relation represents that two (abstract) opens are disjoint from each other and whenever and cover the whole space.
For example, for any bitopological space , we have a tuple where , and, for :
If we only have and , we cannot directly express that a pair is maximal with the property that . I spent quite a lot time finding the right pointfree axiomatisation of maximal pairs. It can be shown that they are characterised by the following two axioms (see Section 4.4.1 of my thesis ):
- (K+) if then
- (K−) if then
Observe that is equivalent to saying that for . So (K+) says that if covers then it must also cover .
These two axioms allowed me to prove everything I needed in my thesis. However, there was one thing bugging me that I wasn’t able to show. Given an arbitrary pair , I wasn’t able to find a pair of opens satisfying (K+) and (K−) while representing the same compact convex subset. In other words, I didn’t know how to calculate for in terms of and alone.
I managed to do it today, three years after submitting my thesis. I would like to thank Guillaume Massas for showing me how to do a special case of this, which prompted me to revisit this topic.
The tuples are called d-frames and are required to satisfy a number of axioms. I will not spell them out here, they are all fairly natural (see my thesis or ). As we said earlier our results apply to compact regular bitopological spaces. Compactness and regularity can be expressed entirely in terms of and :
- For , set to be the largest such that .
- For , set if .
( for and for for are defined symmetrically)
- is regular if for every or .
- is compact if whenever is in then is in , for some finite .
Note that , for in a bitopological space, precisely whenever the -closure of is a subset of . Details and further explanation of these definitions can be found in Sections 2.1.1 and 2.3.1 of my thesis . In the following we assume that is compact and regular.
By denote the set of pairs that satisfy (K+) and (K−). We wish to show the following theorem:
For every there is some such that
- For every and ,
Observe that, if we instantiate with a bitopological space, the two conditions (1) and (2) exactly express that and represent (the complement of) the same convex compact subset.
Fix and define as follows
Observe that this definition yields precisely the pair for in a compact regular bitopological space . Indeed, by the Hofmann-Mislove theorem, the compact downset can be uniquely represented as a Scott-open filter on . Further, is order-isomorphic to the poset of Scott-open filters on (Lemma 5.3.6 in ), where a Scott-open filter is send to the -open . Because implies and because , we obtain the expected equality .
Next we show that is indeed in and that (1) and (2) hold. Showing (1) is immediate, let then, by definition, and because is upwards closed also , giving us . Finally, be regularity, . Symmetrically we also obtain .
We show (2) in two steps.
By compactness and directedness of , there is some such that and . Since it must be that (by the so-called (-) axiom of d-frames). Lastly, because is upwards closed and by (1), from we obtain that .
By compactness for some such that . Because is upwards closed both and are in . However, and by regularity and compactness, satisfies cut rules (see Proposition 6.13 in ), hence .
The reverse implication, entailing from , follows from (1) and the fact that is upwards closed.
Lastly, we explain why . We only check (K+) as (K−) is proved symmetrically. Assume that . From the previous two lemmas we entail that . By compactness and regularity, there exists an such that and . Because and is upwards closed . Therefore, . Finally, gives us that , which implies because is upwards closed. This finishes the proof of Theorem 1.
Observe that item 2 of Theorem 1 guarantees that the choice of is unique. Indeed, for any other pair which satisfies item 2 of Theorem 1, let . From we also have which, by two applications of item 2 of Theorem 1 and (K−), gives . Hence . By regularity . The converse direction and also the equivalent statements for the minus side are proved analogously, hence .
- Achim Jung and M. Andrew Moshier. On the bitopological nature of Stone duality. Technical report, 2006.
- Tomáš Jakl. d-Frames as algebraic duals of bitopological spaces. Charles University and University of Birmingham (Ph.D. thesis), 2018.