Last time, we talked about quasicontinuous domains, locally finitary compact spaces and their Stone duals, the hypercontinuous distributive complete lattices.
From time to time, we happen to discover that several distinct notions are in fact the same, and this is exactly what happened in 20132014, in two papers that appeared about at the same time, with similar discoveries. One is due to Jimmie Lawson and Xiaoyong Xi [3], the other one is due to Achim Jung and myself [2].
I’ll give an introduction to each of the various kinds of spaces that play a role here, separately, and I’ll conclude by telling you… that they all coincide.
QRBdomains
I invented QRBdomains as a quasicontinuous imitation of RBdomains during the summer of 2009, and published a few results about them at the LICS (Logics in Computer Science) conference in 2010 [4], then later in the LMCS (Logical Methods in Computer Science) journal in 2012 [5].
Here is the idea. Among the several equivalent definitions of an RBdomain, one is: X is an RBdomain if and only if there is a directed family of deflations (f_{i})_{i}_{ in }_{I} whose sup is the identity on X (see Theorem 9.6.22 in the book). A deflation is a continuous map from X to X, below the identity, and with finite image. We know that f_{i}(x) is waybelow x for every x in X, so every RBdomain is a continuous dcpo.
What if we required every point to be approximated, not by points f_{i}(x), but by finitary compacts depending continuously on x? The idea is the same as when we went from continuous to quasicontinuous posets.
Formally, let Q(X) denote the Smyth powerdomain of X, consisting of compact saturated subsets ordered by reverse inclusion (see Proposition 8.3.25 in the book). Let
Q_{fin}(X) be its subspace of finitary compacts. We call quasideflation any map φ from X to Q_{fin}(X) that is continuous, has finite image, and is “below the identity” in the sense that x is in φ(x) for every x in X. Alternatively, write η(x) for ↑x; η is a continuous map from X to Q_{fin}(X), and φ is “below the identity” in that new sense if and only if it is below η in the pointwise ordering.
Now call X a QRBdomain if and only if there is a directed family of quasideflations (φ_{i})_{i}_{ in }_{I} on X whose supremum is η. Explicitly, we require that ⋂_{i}_{ in }_{I} φ_{i} (x)=↑x for every x in X.
Unsurprisingly, QRBdomains are quasicontinuous domains. However, they are more than that: they are all stably compact.
Among the properties defining stable compactness, all are satisfied by quasicontinuous domains except two: compactness and coherence. So QRBdomains are pretty wellbehaved: quasicontinuous domains that are also compact and coherent.
The dcpo N_{2} of Figure 5.1, which I have already mentioned in my last post on quasicontinuous dcpo, is a QRBdomain, for example. It is also algebraic; but it is not an RBdomain.
Incidentally, QRBdomains have a rich theory, which parallels that of RBdomains. Notably, I proved [4, 5] that the QRBdomains are exactly the quasiretracts of bifinite domains (provided all domains are secondcountable, see below). Compare this with the fact that the RBdomains are exactly the retracts of bifinite domains. I won’t define what quasiretracts are, except that they are “retracts in the quasi sense”, and that a variant of this result says that the secondcountable QRBdomains are exactly the images of secondcountable bifinite domains by proper maps [5]. (Every proper map defines a canonical kind of quasiretraction, which I called quasiprojection at some point to stress the parallel with projections in the theory of bifinite domains.)
One open question here is whether secondcountability is needed here. I would hope that it is not the case, and would conjecture that every QRBdomain is the quasiretract of a bifinite domain.
I did all this as an attempt to solve a conjecture in probabilistic domain theory [7]. One distinguishing feature is that the probabilistic powerdomain of a QRBdomain is again a QRBdomain, as I’ve shown in [4, 5] in the secondcountable case, and in [2], with Achim Jung, in the general case. The probabilistic powerdomain is a domain of socalled continuous valuations, a variant of the notion of probability measure, and was introduced by Claire Jones in her PhD thesis [6].
QFSdomains
Independently, and by a similar intellectual process, Li and Xu invented a quasicontinuous imitation of FSdomains, see [1].
This really works as for QRBdomains. Say that a map φ : X → Q(X) is quasifinitely separated from the identity (in short, qfs) if and only if there is a finite subset M of X such that for every x in X, there is an m in M such that x ∈ ↑m ⊆ φ(x). A QFSdomain is a dcpo X that has a directed family of qfs maps (φ_{i})_{i}_{ in }_{I} (not necessarily quasideflations) on X whose supremum is η. Again, that means that ⋂_{i}_{ in }_{I} φ_{i} (x)=↑x for every x in X.
Every QRBdomain is a QFSdomain. Li and Xu also showed that every QFSdomain, just like every QRBdomain, is a compact, coherent quasicontinuous dcpo. Is there any QFSdomain that is not a QRBdomain? No. Indeed, we have:
Theorem [2, 3]. The following classes of spaces are the same:

In fact, there are 4 other classes of spaces that also coincide with those, and which may have an interest in their own right, see Theorem 5.7 of [2]. Notably, these spaces are exactly the stably compact spaces that are locally finitary compact (including the fact that their topology must be the Scott topology).
It is time we talked about the proof, and about spaces of kind 3.
Compact, coherent quasicontinuous domains
To prove the theorem, remember that we already know that every QRBdomain is a QFSdomain, and that every QFSdomain is a compact, coherent quasicontinuous domain.
We also know that every quasicontinuous domain is a locally finitary compact space. To put the final nail on the proof of the theorem, we show that every stably compact, locally finitary compact space X is a QRBdomain. This is Proposition 5.6 of [2].
Concretely, fix a stably compact, locally finitary compact space X. We build quasideflations φ_{i} on X, and show that we have enough of them so as to ensure that ⋂_{i}_{ in }_{I} φ_{i} (x)=↑x for every x in X. The fact that X has the Scott topology is due to the fact that this already holds for every sober locally finitary compact space.
The preparatory Proposition 5.4 of [2] gives the main idea already, and there is a similar idea used in Proposition 9.5.31 in the book. Fix a finite semilattice M of compact saturated subsets of X, by which we mean a finite collection of compact saturated subsets that is closed under intersection. Define φ_{M} as mapping each x in X to the smallest element Q of M (with respect to inclusion) whose interior contains x. One checks that φ_{M} is a continuous map from X to Q(X), with finite image. The family of these maps, when M varies, is directed, and this uses the fact that X is compact and coherent. Moreover, the sup of these maps φ_{M} is η, as desired.
Let us check the latter. We need to show that ⋂_{M} φ_{M} (x)=↑x for every x in X, where M ranges over the finite semilattices of compact saturated subsets of X. The righthand side is always included in the lefthand side. To show the converse inclusion, since X is T_{0}, it suffices to show that every open neighborhood U of x contains φ_{M} (x) for some M; it suffices to find a compact saturated neighborhood Q of x included in U (by local compactness), and to take M = {Q, X}. Indeed, for this choice of Q, φ_{M} (x) is equal to Q, which is contained in U.
This looks like a proof, but a crucial ingredient is missing: φ_{M} takes its values in Q(X), not Q_{fin}(X). So this does not prove that X is a QRBdomain. Also, we have not used the (crucial) assumption that X is locally finitary compact.
Instead, we massage each φ_{M} to a new continuous map, replacing each compact saturated subset Q in its image by a slightly larger finitary compact subset neighborhood, using local finitary compactness. This requires some care, notably to keep φ_{M} monotonic, but the fact that φ_{M} has finite image allows us to use induction to ensure this. The argument is not difficult; see the proof of Proposition 5.6 in [2]. (In Appendix A, I’ll discuss why a slightly simpler argument does not work.)
Once we have done so, the problem is fixed, and our arbitrary stably compact, locally finitary compact space X has been revealed as a QRBdomain: our proof is finished.
I would like to state an anecdote, showing how brilliant Achim Jung is. He came to my place (LSV) for a onemonth visit from midMarch to midApril 2013. We discussed a few scientific questions we might explore while he was here, and he expressed the desire to learn about my QRBdomains. After I had explained the basic theory, he walked up to the whiteboard. In his extremely mildmannered way, he said “Now, Jean, tell me if I am wrong, but we can certainly do this […] and that […]”. In a few minutes, he had found the essential argument that I’ve described above. That is how Achim learned the theory… by proving a new, fundamental result about it in a matter of minutes.
[2] Jean GoubaultLarrecq and Achim Jung. QRB, QFS, and the Probabilistic Powerdomain. Proceedings of the 30th Intl. Conf. on Mathematical Foundations of Programming Semantics, ENTCS, pp. 170185, 2014.
[3] Jimmie Lawson and Xiaoyong Xi. The equivalence of QRB, QFS, and compactness for quasicontinuous domains. ORDER, 2014. DOI 10.1007/s1108301493277.
[4] Jean GoubaultLarrecq. ωQRBDomains and the Probabilistic Powerdomain. In LICS’10, pages 352361. IEEE Computer Society Press, 2010.
[5] Jean GoubaultLarrecq. QRBDomains and the Probabilistic Powerdomain. Logical Methods in Computer Science 8(1:14), 2012.
[6] Claire Jones. Probabilistic NonDeterminism. Ph.D. thesis, University of Edinburgh (1990), technical Report ECSLFCS90105.
[7] Achim Jung and Regina Tix. The troublesome probabilistic powerdomain. In: A. Edalat, A. Jung, K. Keimel and M. Kwiatkowska, editors, Proc. 3rd Workshop on Computation and Approximation, Electronic Lecture Notes in Computer Science 13 (1998), pp. 70–91, 23pp.
Appendix A
Here is another idea for proving that the stably compact, locally finitary compact space X is a QRBdomain. This is an idea we explored, and dismissed, because it fails, and I’ll explain why.
Recall that φ_{M} maps every point x in X to the smallest Q in M that is a neighborhood of x. We lamented ourselves that φ_{M}(x) was a compact saturated subset, not necessarily a finitary compact subset of X, and we fixed that by modifying φ_{M} in a second step, using local finitary compactness.
It would seem much simpler to do everything in just one step, and for this the cure seems obvious: only allow M to range over finite semilattices of finitary compact subsets, not general compact saturated subsets. This way, φ_{M} (x) would automatically be in Q_{fin}(X).
This does not work, or at least not in any way that would be simpler. The problem lies in showing that the family of maps φ_{M}, where M now ranges over the semilattices of finitary compact subsets, is directed. In our previous proof, where M was allowed to be any finite semilattice of compact saturated subsets, I said that directedness relied on the fact that X was compact and coherent. Indeed, to find a map φ_{M”} above φ_{M} and φ_{M’}, it sufficed to define M” as the collection of pairwise intersections Q ∩ Q’, Q in M, Q’ in M’. These intersections were compact saturated because X is coherent.
In our new setting, Q and Q’ are now finitary compact… but the best thing we can say about the intersection Q ∩ Q’ is that it is compact saturated, not finitary compact! We could proceed if we could assume that Q ∩ Q’ is finitary compact. Doing so, we would show that every stably compact, locally finitary compact space satisfying the property “every finite intersection of locally finitary compact subsets is locally finitary compact” is a QRBdomain.
However, the additional property fails in some stably compact, locally finitary spaces. In fact, it already fails for some QRBdomains. One example is the space of all triples (a,b,c) in [0, 1]^{3} such that a+b+c≤1, with the ordering defined by (a,b,c) ≤ (a’,b’,c’) if and only if c≤c’, a+b≤a’+b’, b+c≤b’+c’, and a+b+c≤a’+b’+c’. I claim that X is a QRBdomain that does not satisfy the extra condition. The proof is not exactly elementary, but here it goes. X is the space of all subprobability valuations on the threeelement domain {1, 2, ⊤} with 1 and 2 incomparable and below ⊤, encoded as a δ_{1} + b δ_{⊤} + c δ_{2}, where δ_{x} is the Dirac point mass at x. It has been wellknown since Jones’ PhD thesis [6] that δ_{1}=(1,0,0) and δ_{2}=(0,0,1) have uncountably many minimal upper bounds in X, showing that the extra condition fails. X is a QRBdomain because it is the probabilistic powerspace of a QRBdomain [2, 5]; in fact it is even an FSdomain [7].