QRB-Domains and the Probabilistic Powerdomain

Is there any Cartesian-closed category of continuous domains that would be closed under Jones and Plotkin's probabilistic powerdomain construction? This is a major open problem in the area of denotational semantics of probabilistic higher-order languages. We relax the question, and look for quasi-continuous dcpos instead. We introduce a natural class of such quasi-continuous dcpos, the omega-QRB-domains. We show that they form a category omega-QRB with pleasing properties: omega-QRB is closed under the probabilistic powerdomain functor, under finite products, under taking bilimits of expanding sequences, under retracts, and even under so-called quasi-retracts. But... omega-QRB is not Cartesian closed. We conclude by showing that the QRB domains are just one half of an FS-domain, merely lacking control.


The Jung-Tix Problem.
A famous open problem in denotational semantics is whether the probabilistic powerdomain V 1 (X) of an FS-domain X is again an FS-domain [JT98], and similarly with RB-domains in lieu of FS-domains. V 1 (X) (resp. V ≤1 (X)) is the dcpo of all continuous probability (resp., subprobability) valuations over X: this construction was introduced by Jones and Plotkin to give a denotational semantics to higher-order probabilistic languages [JP89].
More generally, is there a category of nice enough dcpos that would be Cartesian-closed and closed under V 1 ? We call this the Jung-Tix problem. By "nice enough", we mean nice enough to do any serious mathematics with, e.g., to establish definability or full abstraction results in extensional models of higher-order, probabilistic languages. It is traditional to equate "nice enough" with "continuous", and this is justified by the rich theory of continuous domains [GHK + 03].
However, quasi-continuous dcpos (see [GLS83], or [GHK + 03, III-3]) generalize continuous dcpos and are almost as well-behaved. We propose to widen the scope of the problem, and ask for a category of quasi-continuous dcpos that would be closed under V 1 . We show that, by mimicking the construction of RB-domains [AJ94], with some flavor of "quasi", is drawn just as X itself, x ∈ X, indicative of the weight of x = 0, = 1 3 δ a + 2 3 δ ⊤ E.g., we obtain a category ωQRB of so-called ωQRB-domains that not only has many desired, nice mathematical properties (e.g., it is closed under taking bilimits of expanding sequences, and every ωQRB-domain is stably compact), but is also closed under V 1 .
We failed to solve the Jung-Tix problem: ωQRB is indeed not Cartesian-closed. In spite of this, we believe our contribution to bring some progress towards settling the question, and at least to understand the structure of V 1 (X) better. To appreciate this, recall what is currently known about V 1 . There are two landmark results: V 1 (X) is a continuous dcpo as soon as X is ( [Eda95], building on Jones [JP89]), and V 1 (X) is stably compact (with its weak topology) whenever X is [JT98,AMJK04]. Since then, no significant progress has been made. When it comes to solving the Jung-Tix problem, we must realize that there is little choice: the only known Cartesian-closed categories of (pointed) continuous dcpos that may suit our needs are RB and FS [JT98]. I.e., all other known Cartesian-closed categories of continuous dcpos, e.g., bc-domains or L-domains, are not closed under V 1 . Next, we must recognize that little is known about the (sub)probabilistic powerdomain of an RB or FS-domain. In trying to show that either RB or FS was closed under V 1 , Jung and Tix [JT98] only managed to show that the subprobabilistic powerdomain V ≤1 (X) of a finite tree X was an RB-domain, and that the subprobabilistic powerdomain of a reversed finite tree was an FS-domain. This is still far from the goal.
In fact, we do not know whether V 1 (X) is an RB-domain when X is even the simple poset {⊥, a, b, ⊤} (a and b incomparable, ⊥ ≤ a, b ≤ ⊤, see Figure 1, right)-but it is an FS-domain. For a more complex (arbitrarily chosen) example, take X to be the finite pointed poset of Figure 2 (i): then V 1 (X) and V ≤1 (X) are continuous and stably compact, but not known to be RB-domains or FS-domains (and they are much harder to visualize, too).
No progress seems to have been made on the question since Jung and Tix' 1998 attempt. As part of our results, we show that for every finite pointed poset X, e.g. Figure 2 (i), V 1 (X) is a continuous ωQRB-domain. This is also one of the basic results that we then leverage to show that V 1 (X) is an ωQRB-domain for any ωQRB-domain, in particular every RB-domain, not just every finite pointed poset, X. One may obtain some intuition as to why this should be so, and at the same time give an idea of what (ω)QRB-domains are. Let X be a finite pointed poset. In attempting to show that V 1 (X) is an RB-domain, we are led to study the so-called deflations f : V 1 (X) → V 1 (X), i.e., the continuous maps f with finite range such that f (ν) ≤ ν for every continuous probability valuation ν on X, and we must try to find deflations f such that f (ν) is as close as one desires to ν. All natural definitions of f fail to be continuous, and in fact to be monotonic. (E.g., Graham's construction [Gra88] is not monotonic, see Jung and Tix.) Looking for maps f such that f (ν) is instead a finite, non-empty set of valuations below ν shows more promise-the monotonicity requirements are slightly more relaxed. Such a set-valued function is what we call a quasi-deflation below. For example, one may think of fixing N ≥ 1 (N = 3 in Figure 1), and mapping ν to the collection of all valuations ν ′ below ν such that the measure of any subset is a multiple of 1/N , keeping only those ν ′ that are maximal. (Pick them from the left of Figure 1, in our example.) This still does not provide anything monotonic, but we managed to show that one can indeed approximate every element ν of V 1 (X), continuously in ν, using quasi-deflations. The proof is non-trivial, and rests on deep properties relating QRB-domains and quasi-retractions, all notions that we define and study.
1.2. Outline. We introduce most of the required notions in Section 2. Since we shall only start studying the probabilistic powerdomain in Section 6, we shall refrain from defining valuations, probabilities, and related concepts until then.
We introduce QRB-domains in Section 3. They are defined just as RB-domains are, only with a flavor of "quasi", i.e., replacing approximating elements by approximating sets of elements. We establish their main properties there, in particular that they are quasi-continuous, stably compact, and Lawson-compact. Much as RB-domains are also characterized as the retracts of bifinite domains, we show that, up to a few details, the QRB-domains are the quasi-retracts of bifinite domains in Section 4. This allows us to parenthesize QRB as quasi-(retract of bifinite domain) or as (quasi-retract) of bifinite domain. Quasi-retractions are an essential concept in the study of QRB-domains, and we introduce them here, as well as the related notion of quasi-projections-images by proper maps.
We also show that the category of countably-based QRB-domains is closed under finite products (easy) and taking bilimits of expanding sequences (hard, but similar to the case of RB-domains) in Section 5.
The core of the paper is Section 6, where we show that the category ωQRB of countablybased QRB-domains is closed under the probabilistic powerdomain construction. This capitalizes on all previous sections, and will follow from a variant of Jung and Tix' result that V 1 (X) is an RB-domain whenever X is a finite tree, and applying suitable quasiprojections and bilimits. The key result will then be Theorem 6.5, which shows that for any quasi-projection Y of a stably compact space X, V 1 (Y ) is again a quasi-projection of V 1 (X), again up to a few details.
We conclude in Section 7.
1.3. Other Related Work. Instead of solving the Jung-Tix problem, one may try to circumvent it. One of the most successful such attempts led to the discovery of qcb-spaces [BSS07] and to compactly generated countably-based monotone convergence spaces [BSS06], as Cartesian-closed categories of topological spaces where a reasonable amount of semantics can be done. This provides exciting new perspectives. The category of qcb-spaces accommodates two probabilistic powerdomains [BS09]. The observationally induced one is essentially V 1 (X) (with the weak topology), but differs from the one obtained as a free algebra.

Preliminaries
We refer to [AJ94, GHK + 03, Mis98] for background material. A poset X is a set with a partial ordering ≤. Let ↓ A be the downward closure {x ∈ X | ∃y ∈ A · x ≤ y}; we write ↓ x for ↓{x}, when x ∈ X. The upward closures ↑ A, ↑ x are defined similarly. When x ≤ y, x is below y and y is above x. X is pointed iff it has a least element ⊥. A dcpo is a poset X where every directed family (x i ) i∈I has a least upper bound sup i∈I x i ; directedness means that I = ∅ and for every i, i ′ ∈ I, there is an i ′′ ∈ I such that Every poset, and more generally each preordered set X comes with a topology, whose opens U are the upward closed subsets such that, for every directed family (x i ) i∈I that has a least upper bound in U , x i ∈ U for some i ∈ I. This is the Scott topology. When we see a poset or dcpo X as a topological space, we will implicitly assume the latter, unless marked otherwise.
There is a deep connection between order and topology. Given any topological space X, its specialization preorder ≤ is defined by x ≤ y iff every open containing x also contains y. X is T 0 iff ≤ is an ordering, i.e., x ≤ y and y ≤ x imply x = y. The specialization preorder of a dcpo X (with ordering ≤, and equipped with its Scott topology), is the original ordering ≤.
A subset A of a topological space X is saturated iff it is the intersection of all opens U containing A. Equivalently, A is upward closed in the specialization preorder [ The interior int(A) of a subset A of a topological space X is the largest open contained in A. A is a neighborhood of x if and only if x ∈ int(A), and a neighborhood of a subset B if and only if B ⊆ int(A). A subset Q of a topological space X is compact iff one can extract a finite subcover from every open cover of Q. The important ones are the saturated compacts. X is locally compact iff for each open U and each x ∈ U , there is a compact saturated subset Q such that x ∈ int(Q) and Q ⊆ U . In any locally compact space, we have the following interpolation property: whenever Q is a compact subset of some open U , then there is a compact saturated subset Q 1 such that Q ⊆ int(Q 1 ) ⊆ Q 1 ⊆ U .
X is sober iff every irreducible closed subset is the closure of a unique point; in the presence of local compactness (and when X is T 0 ), it is equivalent to require that X be well-filtered [GHK + 03, Theorem II-1.21], i.e., to require that, for every open U , for every filtered family (Q i ) i∈I of saturated compacts such that ↓ i∈I Q i ⊆ U , Q i ⊆ U for some i ∈ I already. We say that the family is filtered iff it is directed in the ⊇ ordering, and make it explicit by using ↓ as superscript. (Symmetrically, we write ↑ for directed unions.) Given a topological space X, let Q(X) be the collection of all non-empty compact saturated subsets Q of X. There are two prominent topologies one can put on Q(X). The upper Vietoris topology has a subbase of opens of the form ✷U , U open in X, where we write ✷U for the collection of compact saturated subsets Q ′ included in U . We shall write Q V (X) for the space Q(X) with the upper Vietoris topology, and call it the Smyth powerspace. The specialization ordering of Q V (X) is reverse inclusion ⊇. On the other hand, we shall reserve the notation Q σ (X) for the Smyth powerdomain of X, which is equipped with the Scott topology of ⊇ instead. When X is well-filtered, Q(X) is a dcpo, with least upper bounds of directed families computed as filtered intersections, and ✷U is Scott-open for every open subset U of X, i.e., the Scott topology is finer than the upper Vietoris topology. When X is locally compact and sober (in particular, well-filtered), the two topologies coincide, and Q σ (X) is then a continuous dcpo (see below), where Q ≪ Q ′ iff Q ′ ⊆ int(Q) [GHK + 03, Proposition I-1.24.2]. Schalk [Sch93, Chapter 7] provides a deep study of these spaces.
For every finite subset E of a topological space X, E is compact and ↑ E is saturated compact in X. We call finitary compact those subsets of the form ↑ E with E finite, and let Fin(X) be the subset of Q(X) consisting of the non-empty finitary compacts. Fin(X) can be topologized with the subspace topology from Q V (X), in which case we obtain a space we write Fin V (X), or with the Scott topology of reverse inclusion ⊇, yielding a space that we write Fin σ (X).
Given any poset X, any finite subset E of X, and any element x of X, we write E ≤ x iff x ∈ ↑ E, i.e., iff there is a y ∈ E such that y ≤ x. Given any upward closed subset U of X, we shall write U Î x iff for every directed family (x i ) i∈I that has a least upper bound above x, then x i is in U for some i ∈ I. Then a finite set E approximates x iff ↑ E Î x. This is usually written E ≪ x in the literature. We shall also write y ≪ x, when y ∈ X, as shorthand for ↑ y Î x. This is the more familiar way-below relation, and a poset is continuous if and only if the set ↓ ↓ x of all elements y such that y ≪ x is directed and has x as least upper bound. One should be aware that ↑ E Î x means that the elements of E approximate x collectively, while none in particular may approximate x individually. E.g., in the poset N 2 (Figure 2 (ii)), the sets {(0, m), (1, n)} approximate ω, for all m, n ∈ N; but (0, m) ≪ ω, (1, n) ≪ ω.
It may be helpful to realize that Fin(X) can also be presented in the following equivalent way. Given two finitary compacts ↑ E and ↑ E ′ , ↑ E ⊇ ↑ E ′ if and only if for every x ′ ∈ E ′ , there is an x ∈ E such that x ≤ x ′ , and then we write E ≤ ♯ E ′ : this is the so-called Smyth preorder . Then we can equate the finitary compacts ↑ E with the equivalence classes of finite subsets E, up to the equivalence ≡ defined by For our purposes, an RB-domain is a pointed dcpo X with a directed family (f i ) i∈I of deflations such that sup i∈I f i = id X [AJ94, Exercise 4.3.11(9)]. A deflation f on X is a continuous map from X to X such that f (x) ≤ x for every x ∈ X, and that has finite image. We order deflations, as well as all maps with codomain a poset, pointwise: i.e., f ≤ g iff f (x) ≤ g(x) for every x ∈ X; knowing this, directed families and least upper bounds of deflations make sense. Every RB-domain is a continuous dcpo, and f i (x) ≪ x for every i ∈ I and every x ∈ X.
A B-domain is defined similarly, except the deflations f i are now required to be idem- An FS-domain is defined similarly again, except the functions f i are no longer deflations, but continuous functions that are finitely separated from id X . That is, we now require that there is a finite set M i such that for every x ∈ X, there is an m ∈ M i such that f i (x) ≤ m ≤ x. We say that M i is finitely separating for f i on X.
Every deflation is finitely separated from id X : take M i to be the image of f i . The converse fails. E.g., for every ǫ > 0, the function x → max(x − ǫ, 0) is finitely separated from the identity on [0, 1], but is not a deflation [JT98, Section 3.2]. Every RB-domain is an FS-domain. The converse is not known.
A quasi-continuous dcpo X (see [GLS83] or [GHK + 03, Definition III-3.2]) is a dcpo such that, for every x ∈ X, the collection of all ↑ E ∈ Fin(X) that approximate The theory of quasi-continuous dcpos is less well explored than that of continuous dcpos, but quasi-continuous dcpos retain many of the properties of the latter. (Every continuous dcpo is quasi-continuous, but not conversely. A counterexample is given by N 2 , see Figure 2 (ii).) Every quasi-continuous dcpo X is locally compact and sober in its Scott topology [GHK + 03, III-3.7]. In a quasi-continuous dcpo X, for every ↑ E ∈ Fin(X), the set ↑ ↑ E defined as {x ∈ X | ↑ E Î x}, is open, and equals the interior int(↑ E) [GHK + 03, III-3.6(ii)]; every open U is the union of all the subsets ↑ ↑ E with ↑ E ∈ Fin(X) contained in U [GHK + 03, III-5.6]; and for every compact saturated subset Q and every open subset U containing Q, there is a finitary compact subset ↑ E of X such that Q ⊆ ↑ ↑ E and ↑ E ⊆ U [GHK + 03, III-5.7]. In particular, Q = ↓ ↑ E∈Fin(X), Q⊆↑ ↑ E ↑ E. Another consequence is interpolation: quasi-continuous dcpo X, for some ↑ E ∈ Fin(X), and x ∈ X, then ↑ E Î ↑ E ′ Î x for some ↑ E ′ ∈ Fin(X). If X is a quasi-continuous dcpo, the formula Q = ↓ ↑ E∈Fin(X), Q⊆↑ ↑ E ↑ E, valid for every Q ∈ Q(X), shows that Q is the filtered intersection of its finitary compact neighborhoods, equivalently the directed least upper bound of those non-empty finitary compacts ↑ E (E ∈ Fin(X)) that are way-below Q. In other words, the finitary compacts form a basis of Q(X).

QRB-Domains
We model QRB-domains after RB-domains, replacing single approximating elements f i (x), where f i is a deflation, by finite subsets, as in quasi-continuous dcpos.
Definition 3.1 (QRB-Domain). A quasi-deflation on a poset X is a continuous map ϕ : X → Fin σ (X) such that x ∈ ϕ(x) for every x ∈ X, and im ϕ = {ϕ(x) | x ∈ X} is finite. A QRB-domain is a pointed dcpo X with a generating family of quasi-deflations, i.e., a directed family of quasi-deflations (ϕ i ) i∈I with ↑ x = ↓ i∈I ϕ i (x) for each x ∈ X. We order quasi-deflations pointwise, i.e., ϕ ≤ ψ iff ϕ(x) ⊇ ψ(x) for every x ∈ X. Above, we write ↓ instead of to stress the fact that the family (ϕ i (x)) i∈I of which we are taking the intersection is filtered, i.e., for any two i, i ′ ∈ I, there is an i ′′ ∈ I such that ϕ i ′′ (x) is contained in both ϕ i (x) and ϕ i ′ (x). It is equivalent to say that (ϕ i (x)) i∈I is directed in the ⊇ ordering of Fin(X).
One can see the finitary compacts ϕ i (x) as being smaller and smaller upward closed sets containing x. The intersection ↓ i∈I ϕ i (x) is then just the least upper bound of (ϕ i (x)) i∈I in the Smyth powerdomain Q(X). On the other hand, X embeds into Q V (X) by equating x ∈ X with ↑ x ∈ Q(X). Modulo this identification, the condition ↑ x = ↓ i∈I ϕ i (x) requires that x is the least upper bound of (ϕ i (x)) i∈I in Q(X).
That ϕ is continuous means that ϕ is monotonic (x ≤ y implies ϕ(x) ⊇ ϕ(y)), and that for every directed family (x j ) j∈J of elements of X, ϕ(sup j∈J x j ) is equal to ↓ i∈I ϕ(x j )-this implies that the latter is finitary compact, in particular.
is the set of upper bounds of (f i (x)) i∈I , of which the least is x. So this set is exactly ↑ x.
We shall improve on this in Theorem 7.3, which implies that not only the RB-domains, but all FS-domains, are QRB-domains.
For any deflation f , and more generally whenever f is finitely separated from the identity, f (x) is way-below x [GHK + 03, Lemma II-2.16]. Similarly: Lemma 3.3. Let X be a poset, and ϕ be a quasi-deflation on X. For every x ∈ X, ϕ(x) Î x.
RB-domains, and more generally FS-domains, are not just continuous domains, they are stably compact, i.e., locally compact, sober, compact and coherent (see, e.g., [AJ94, Theorem 4.2.18]). We say that a topological space is coherent iff the intersection of any two compact saturated subsets is compact (and saturated). In a stably compact space, the intersection of any family of compact saturated subsets is compact. We show that QRB-domains are stably compact as well.
Since every quasi-continuous dcpo is locally compact and sober [GHK + 03, Proposition III-3.7], and also compact since pointed, only coherence remains to be shown. For this, we need the following consequence of Rudin's Lemma, a finitary form of well-filteredness: It follows that, if X is a dcpo, then the Scott topology on Fin(X) is finer than the upper Vietoris topology. Indeed, this reduces to showing that Fin(X) ∩ ✷U is Scott-open in Fin(X), for every open subset U of X. And this is Proposition 3.5, plus the easily checked fact that ✷U is upward closed in ⊇.
Corollary 3.6. Let X be a dcpo. The Scott topology is finer than the upper Vietoris topology on Fin(X), and coincides with it whenever X is quasi-continuous.
also form a directed family in Fin(X), and their intersection is ↑ E. So there are finitary compacts Chapter 7] proved that Q V defines a monad on the category of topology spaces (see [Mog91] for an introduction to monads and their importance in programming language semantics). This means first that there is a unit map η X -here, η X maps x ∈ X to ↑ x ∈ Q V (X), and this is continuous because η −1 X (✷U ) = U . That Q V is a monad also means that every continuous map h : Again, h † is continuous, because h † −1 (✷U ) = ✷h −1 (✷U ). And the monad laws are satisfied: One should be careful here: Q V is a monad, but Q σ is not a monad, except on specific subcategories, e.g., sober locally compact spaces X, where Q σ (X) = Q V (X) anyway.
The continuity claims in the following lemma are then obvious.
Lemma 3.7. Let X, Y be topological spaces. Given any continuous map ψ : Proof. In each case, one only needs to show that ψ † maps relevant compacts to finitary compacts. In the first case, for every finitary compact , and this is finitary compact. In the second case, ψ † (Q) = x∈Q ψ(x) is a finite union of finitary compacts since im ψ is finite.
One would also like ψ † to be continuous from Q σ (X) to Fin σ (Y ), in the face of the importance of the Scott topology. This is a consequence of the above when X is sober and locally compact, and Y is a quasi-continuous dcpo, since Q σ (X) = Q V (X) and Fin σ (Y ) = Fin V (Y ) in this case. However, one can also prove this in a more general setting, using the following observation. For each topological space Z, write Z σ for Z with the Scott topology of its specialization preorder. For short, we shall call quasi monotone convergence space any space Z such that the (Scott) topology on Z σ is finer than that of Z, i.e., such that every open subset of Z is open is Scott-open. This is a slight relaxation of the notion of monotone convergence space, i.e., of a quasi monotone convergence space that is a dcpo in its specialization preorder [GHK + 03, Definition II-3.12]. E.g., every sober space is a monotone convergence space, and in particular a quasi monotone convergence space.
Lemma 3.8. Let Z be a quasi monotone convergence space and Z ′ be a topological space.
Proof. Since f is continuous, it is monotonic with respect to the underlying specialization preorders. Let (z i ) i∈I be any directed family of elements of Z, with least upper bound z. Certainly f (z) is an upper bound of (f (z i )) i∈I . Let us show that, for any other upper bound When X is sober and locally compact, the topology of Q σ (X) coincides with that of Q V (X). In particular, Z = Q V (X) is a quasi-monotone convergence space. Taking Z ′ = Q V (Y ) in Lemma 3.8, one obtains the following corollary.
Corollary 3.9. Let X be a sober, locally compact space, and Y be a topological space. Every Corollary 3.10. Let Y be a topological space, Z be a quasi monotone convergence space. Every continuous map from Z to Fin V (Y ) is Scott-continuous, i.e., continuous from Z σ to Fin σ (Y ).
Lemma 3.11. Let X be a QRB-domain, and (ϕ i ) i∈I a generating family of quasi-deflations. For every open subset U of X, ↑ i∈I ϕ −1 Lemma 3.12. Let X be a QRB-domain, and (ϕ i ) i∈I a generating family of quasi-deflations.
. Since X is quasi-continuous (Corollary 3.4), it is sober and locally compact. So Corollary 3.9 applies, showing that ϕ † i is Scott-continuous from Q(X) to Q(X). Lemma 3.12 states that the least upper bound of (ϕ † i ) i∈I is the identity on Q(X). Clearly, ϕ † i has finite image. So Q(X) is an RB-domain. Theorem 3.14. Every QRB-domain is stably compact.
Proof. Let X be a QRB-domain, with generating family of quasi-deflations (ϕ i ) i∈I . We claim that, given any two compact saturated subsets Q and Q ′ of X, Q∩Q ′ is again compact saturated. This is obvious if Q∩Q ′ is empty. Otherwise, writing ↑ Y y for the upward closure of an element y of a poset Y , ↑ Q(X) Q∩↑ Q(X) Q ′ is an intersection of two finitary compacts in Q V (X). Since X is a quasi-continuous dcpo by Corollary 3.4, X is sober and locally compact, the left to right inclusion is obvious, and conversely every X is compact since pointed, and also locally compact and sober, as a quasi-continuous dcpo, hence stably compact.
The Lawson topology is the smallest topology containing both the Scott-opens and the complements of all finitary compacts ↑ E ∈ Fin(X). When X is a quasi-continuous dcpo, since ↑ E is compact saturated and every non-empty compact saturated subset is a filtered intersection of such sets ↑ E, the Lawson topology coincides with the patch topology, i.e., the smallest topology containing the original Scott topology and all complements of compact saturated subsets. Every stably compact space is patch-compact, i.e., compact in its patch topology [GHK + 03, Section VI-6]. So: In the sequel, we shall need some form of countability: Definition 3.16. An ωQRB-domain is a QRB-domain with a countable generating family of quasi-deflations.
for every x ∈ X. Proof. Let X be an ωQRB-domain, and (ψ j ) j∈N be a countable generating family of quasideflations. Build a sequence (j i ) i∈N by letting j 0 = 0, and j i+1 be any j ∈ N such that ψ j is above ψ i and ψ j i , by directedness. Then let ϕ i = ψ j i for every i ∈ N. By construction, Recall that a topological space is countably-based if and only if it has a countable subbase, or equivalently, a countable base.
We claim that the countably many subsets int(ϕ i (y)), y ∈ E j , i, j ∈ N, form a base of the topology.
It is enough to show that, for every open U and every element x ∈ U , x ∈ int(ϕ i (y)) for some y ∈ E j , i, j ∈ N, such that ϕ i (y) ⊆ U : since ↑ x = ↓ j∈N ϕ j (x) ⊆ U , use Proposition 3.5 to find j ∈ N such that ϕ j (x) ⊆ U . Since x ∈ ϕ j (x) and ϕ j (x) = ↑ E jk for some k, there is a y ∈ E jk ⊆ E j such that y ≤ x, and y ∈ U . Repeating the argument on y, we find i ∈ N such that ϕ i (y) ⊆ U . By Lemma 3.3, ϕ i (y) Î y, i.e., y is in int(ϕ i (y)) since X is quasi-continuous. Since y ≤ x, x is in int(ϕ i (y)).
If: let (ϕ i ) i∈I be a generating family of quasi-deflations on X, and assume that the topology of X has a countable base {U k | k ∈ N}. Assume without loss of generality that U k = ∅ for every k ∈ N. For every pair ℓ, k ∈ N such that U ℓ ⊆ ↑ E ⊆ U k for some finite set E, pick one such finite set and call it E ℓk . One can enumerate all such pairs as ℓ m , k m , m ∈ N. By Lemma 3.12, pick such an i and call it i m . By directedness, we may also assume that ϕ im is also above ϕ in , 0 ≤ n < m. Define ψ m as ϕ im . This yields a non-decreasing sequence of quasi-deflations (ψ m ) m∈N .
We claim that it is generating. The RB-domains can be characterized as the retracts of bifinite domains. Recall that a retraction of X onto Y is a continuous map r : X → Y such that there is continuous map s : Y → X (the section) with r(s(y)) = y for every y ∈ Y .
We shall show that (ω)QRB-domains are not just closed under retractions, but under a more relaxed notion that we shall call quasi-retractions. More precisely, our aim in this section is to show that the ωQRB-domains are exactly the quasi-retracts of bifinite domains, up to some details.
For each continuous r : Qr is continuous, since Qr −1 (✷V ) = ✷r −1 (V ) for every open V . This is the action of the Q V functor of the Smyth powerspace monad [Sch93, Chapter 7], equivalently In diagram notation, we require the bottom right triangle to commute, but not the top left triangle-what the puncture + indicates; the outer square always commutes: While a section s : Y → X picks an element s(y) in the inverse image r −1 (y), continuously, a quasi-section is only required to pick a non-empty compact saturated collection of elements from r −1 (↑ y) meeting r −1 (y) (see Figure 3), continuously again.
The converse fails. For example, N 2 is a quasi-retract of N ω + N ω (see Figure 2 (iii)): r maps both (0, ω) and (1, ω) to ω ∈ N 2 , and ς(y) = r −1 (↑ y) for every y. But Y is not a retract of X: X is a continuous dcpo, and every retract of a continuous dcpo is again one; recall that N 2 is not continuous.
Every quasi-retraction r : X → Y induces a continuous map η Y •r : X → Q V (Y ), which is then a retraction in the Kleisli category C C C Q . A retraction in a category is a morphism r : X → Y such that there is a section morphism s : Y → X, i.e., one with r • s = id Y . It is easy to see that the quasi-retractions are exactly those continuous maps r : Lemma 4.2. Every quasi-retraction r : X → Y onto a T 0 space Y is surjective. More precisely, if ς is a matching quasi-section, then every element y ∈ Y is of the form r(x) for some x ∈ ς(y).
The following is reminiscent of the fact that every retract of a stably compact space is again stably compact [Law87, Proposition, bottom of p.153, and subsequent discussion]: we shall show that any T 0 quasi-retract of a stably compact space is stably compact. We start with compactness.
Proof. The image of a compact set by a continuous map is compact. Now apply Lemma 4.2.
Lemma 4.4. Any quasi-retract Y of a well-filtered space X is well-filtered.
Proof. Let r : X → Y be the quasi-retraction, with quasi-section ς : Y → Q V (X).
Let (Q i ) i∈I be a filtered family of compact saturated subsets of Y , and assume that . This is compact saturated, and forms a directed family, since ς † is monotonic. We claim that i∈I Q ′ i ⊆ r −1 (V ). Indeed, every x ∈ i∈I Q ′ i is such that, for every i ∈ I, there is a y i ∈ Q i such that x ∈ ς(y i ); then r(x) ∈ Qr(ς(y i )) = ↑ y i , so r(x) ∈ Q i , for every i ∈ I.
Lemma 4.6. Any quasi-retract Y of a locally compact space X is locally compact.
Proof. Let r : X → Y be the quasi-retraction, with quasi-section ς : Y → Q V (X). Let y be any point of Y , and V be an open neighborhood of y. Since y ∈ V , Qr(ς(y)) = ↑ y ⊆ V , so ς(y) ⊆ r −1 (V ). Observe that ς(y) is compact saturated and r −1 (V ) is open in X. Use interpolation in the locally compact space X: there is a compact saturated subset Q 1 such that ς(y) ⊆ int(Q 1 ) ⊆ Q 1 ⊆ r −1 (V ).
In particular, ς(y) ∈ ✷int(Q 1 ), so y is in the open subset ς −1 (✷int(Q 1 )). The latter is included in the compact subset Qr(Q 1 ), since every element y ′ of it is such that ς(y ′ ) ⊆ int(Q 1 ) ⊆ Q 1 , hence ↑ y ′ = Qr(ς(y ′ )) ⊆ Qr(Q 1 ). In particular, y is in the interior of Proposition 4.7. Every T 0 quasi-retract Y of a stably compact space X is stably compact.
Proof. Y is T 0 by assumption, and locally compact, well-filtered, compact, and coherent by Lemma 4.3, Lemma 4.4, Lemma 4.5, and Lemma 4.6. In the presence of local compactness, it is equivalent to require sobriety or to require the space to be T 0 and well-filtered [GHK + 03, Theorem II-1.21].
Call a space X locally finitary if and only if for every x ∈ X and every open neighborhood U of x, there is a finitary compact ↑ E such that x ∈ int(↑ E) and ↑ E ⊆ U . This is the same definition as for local compactness, replacing compact saturated subsets by finitary compacts. The interpolation property of locally compact spaces refines to the following: In a locally finitary space X, if Q is compact saturated and included in some open subset U , then there is a finitary compact ↑ E such that Q ⊆ int(↑ E) and ↑ E ⊆ U . The proof is as for interpolation in locally compact spaces: for each x ∈ Q, pick a finitary compact ↑ E x such that x ∈ int(↑ E x ) and ↑ E x ⊆ U . (int(↑ E x )) x∈Q is an open cover of Q. Since Q is compact, it has a finite subcover ↑ E 1 , . . . , ↑ E n . Then take E = E 1 ∪ . . . ∪ E n .
We observe right away the following analog of Lemma 4.6.
Lemma 4.8. Any quasi-retract Y of a locally finitary space X is locally finitary.
Proof. As in the proof of Lemma 4.6, let y ∈ Y and V be an open neighborhood of y. By interpolation between Q = ς(y) and U = r −1 (V ) in the locally finitary space X, we find a finitary compact subset Q 1 = ↑ E 1 of X such that ς(y) ⊆ int(Q 1 ) ⊆ Q 1 ⊆ r −1 (V ). The rest of the proof is as for Lemma 4.6, only noticing that Qr(Q 1 ) = ↑ r(E 1 ) is finitary compact.
The importance of locally finitary spaces lies in the following result: see Banaschewski [Ban77], or the equivalence between Items (6) and (11)   We use this, in particular, in the following proposition. Proof. Let X be a QRB-domain, Y be a T 0 space, r : X → Y be a quasi-retraction, and ς : Y → Q V (X) be a matching quasi-section. We first note that Y is stably compact, by Proposition 4.7, using the fact that X is itself stably compact (Theorem 3.14). So Y is sober. By Proposition 4.9, X is locally finitary, so Y is, too, by Lemma 4.8. By Proposition 4.9 again, Y is a quasi-continuous dcpo, and its topology is the Scott topology.
Note that Y is pointed. Letting ⊥ be the least element of X, r(⊥) is the least element of Y : for every y ∈ Y , pick some x ∈ X such that r(x) = y by Lemma 4.2, then r(⊥) ≤ r(x) = y.
Let now (ϕ i ) i∈I be a generating family of quasi-deflations on X. Clearly, if ϕ i is below ϕ j , then ϕ i is below ϕ j , so ( ϕ i ) i∈I is directed.
The case of ωQRB-domains is similar, where now (ϕ i ) i∈N is a generating sequence of quasi-deflations.
Later, we shall need a refinement of the notion of quasi-retraction, which is to the latter as projections are to retractions. Recall that a projection is a retraction r : X → Y , with section s, such that additionally s • r ≤ id X . Similarly, it is tempting to define a quasiprojection as a quasi-retraction (with quasi-section ς) such that x ∈ ς(r(x)) for every x ∈ X. If r is a retraction, with section s, and we see r as a quasi-retraction in the canonical way, defining ς(y) as ↑ s(y), then the quasi-projection condition x ∈ ς(r(x)) is equivalent to the projection condition (s • r)(x) ≤ x.
The point x shown in Figure 3 satisfies the condition x ∈ ς(r(x)): x is in the gray area ς(y), where y = r(x). However, Lemma 4.11 below shows that r is not a quasi-projection: for this to be the case, the gray area ς(y) should fill the whole of r −1 (↑ y).
There is no need to invent a new term, though: Lemma 4.11 shows that quasi-projections are nothing else than proper surjective maps. A map r : X → Y is proper if and only if it is continuous, ↓ r(F ) is closed in Y for every closed subset F of X, and r −1 (↑ y) is compact in X for every element y of Y [GHK + 03, Lemma VI-6.21 (i)].
Lemma 4.11. Let X be a topological space, and Y be a T 0 topological space. For a map r : X → Y , the following two conditions are equivalent: (1) r is a quasi-retraction, with matching quasi-section ς : Y → Q V (X), such that additionally x ∈ ς(r(x)) for every x ∈ X; (2) r is proper and surjective. Then the quasi-section ς in (1) is unique, and it is defined by ς(y) = r −1 (↑ y).
Proof. We first prove the following fact, which will serve in both directions of proof: ( * ) assume ς(y) = r −1 (↑ y) for every y ∈ Y , then for every open subset U of X, the complement of ς −1 (✷U ) in Y is ↓ r(F ), where F is the complement of U in X. Indeed, the complement of ς −1 (✷U ) is the set of elements y ∈ Y such that ς(y) is not included in U , i.e., such that there is an x ∈ ς(y) that is not in U , i.e., in F . Since ς(y) = r −1 (↑ y), this is the set of elements y such that there is an x ∈ F such that y ≤ r(x), namely, ↓ r(F ).
Assume r is a quasi-retraction, and ς is a matching quasi-section such that x ∈ ς(r(x)) for every x ∈ X. We have seen that r is surjective (Lemma 4.2).
It also follows that r −1 (↑ y) is compact in X. And, using ( * ), for every closed subset F of X, with complement U , ↓ r(F ) is the complement of ς −1 (✷(U )), which is open since ς is continuous, so ↓ r(F ) is closed. Therefore r is proper.
Conversely, assume that r is proper and surjective. Define ς(y) as r −1 (↑ y). Since r is surjective, ς(y) is non-empty. It is saturated, i.e., upward closed, because r is monotonic. Since r −1 (↑ y) is compact, ς(y) is an element of Q(Y ). For every open subset U of X, with complement F , ς −1 (✷U ) is the complement of ↓ r(F ) by ( * ), hence is open since r is proper. So ς is continuous.
Let us turn to bifinite domains, or rather to their countably-based variant. Countability will be needed in a few crucial places.
A pointed dcpo X is an ωB-domain (a.k.a. an SFP-domain) iff there is a non-decreasing sequence of idempotent deflations (f i ) i∈N such that, for every x ∈ X, x = sup i∈N f i (x). I.e., an ωB-domain is just like a B-domain, except that we take a non-decreasing sequence, not a general directed family of idempotent deflations.
The key lemma to prove Theorem 4.13 below is the following refinement of Rudin's Lemma [GHK + 03, III-3.3]. Note that Rudin's Lemma would only secure the existence of a directed family Z whose least upper bound is y, and which intersects each E 0 i ; but Z may intersect each E 0 i in more than one element y i . We pick exactly one element y i in each E 0 i , and for this countability seems to be needed.
Lemma 4.12. Let Y be a dcpo, y ∈ Y , and (↑ E 0 i ) i∈N a non-decreasing sequence in Fin(Y ) (w.r.t. ⊇) such that ↑ y = ↓ i∈N ↑ E 0 i . There is a non-decreasing sequence (y i ) i∈N in Y such that y i ∈ E 0 i for every i ∈ N, and sup i∈N y i = y.
Build a tree as follows. Informally, there is a root node, all (non-root) nodes at distance i ≥ 1 from the root node are labeled by some element of E i−1 , and each such node N , labeled y i−1 , say, has as many successors as there are elements y i in E i such that y i−1 ≤ y i . Formally, one can define the nodes as being the sequences y 0 , y 1 , . . . , y i−1 , i ∈ N, where y 0 ∈ E 0 , y 1 ∈ E 1 , . . . , y i−1 ∈ E i−1 , and y 0 ≤ y 1 ≤ . . . ≤ y i−1 . Such a node is labeled y i−1 (if i ≥ 1), and its successors are all the sequences y 0 , y 1 , . . . , y i−1 , y i with y i chosen in E i , and above This tree has arbitrarily long branches (paths from the root). Indeed, for every i ∈ N, pick an element y i ∈ E i -this is possible since y ∈ ↑ E i , hence E i is non-empty-, then an element y i−1 ∈ E i−1 below y i -since ↑ E i−1 ⊇ ↑ E i -, then an element y i−2 ∈ E i−2 below y i−1 , . . . , and finally an element y 0 ∈ E 0 below y 1 . This is a node at distance i + 1 from the root.
It follows that the tree is infinite. It is finitely-branching, meaning that every node has only finitely many successors-because E i is finite. Kőnig's Lemma then states that this tree must have an infinite branch. Reading the labels on non-root nodes in this branch, we obtain an infinite sequence  (ii) ⇒ (i). Write Y as a quasi-retract of an ωB-domain X. X is trivially an ωQRBdomain. Since Y , as a dcpo, is T 0 , Proposition 4.10 applies, so Y is an ωQRB-domain.
(i) ⇒ (iii). Let Y be an ωQRB-domain, with generating sequence of quasi-deflations (ϕ i ) i∈N . Let im ϕ i = {↑ E i1 , . . . , ↑ E in i }, and define E i as the finite set n i j=1 E ij , for each i ∈ N. Let X be the set of all non-decreasing sequences y = (y i ) i∈N in Y such that y i ∈ j≤i E j , and y i ∈ ϕ i (sup k∈N y k ). Order X componentwise. As in [Jun88, Theorem 4.9, Theorem 4.1], X is an ωB-domain: for each i 0 ∈ N, consider the idempotent deflation f i 0 defined by f i 0 ( y) = (y min(i,i 0 ) ) i∈N . To show that this is well-defined, we must show that y min(i,i 0 ) ∈ ϕ i (sup k∈N y min(k,i 0 ) ), i.e., that y min ( Let now r : X → Y map y to sup i∈N y i . This is evidently Scott-continuous. For any fixed y ∈ Y , apply Lemma 4.12 with ↑ E 0 i = ϕ i (y) to obtain a non-decreasing sequence y = (y i ) i∈N such that y i ∈ ϕ i (y) for every i ∈ N and sup i∈N y i = y: in particular, y is in Y , and r( y) = y. So r is surjective. Let us show that it is proper.
To this end, we first remark that r −1 (↑ y) = { y ∈ X | ∀i ∈ N · y i ∈ ϕ i (y)}. Indeed, if y = (y i ) i∈N is in r −1 (↑ y), then y ≤ r( y) = sup k∈N y k , and since y ∈ X, y i ∈ ϕ i (sup k∈N y k ) ⊆ ϕ i (y), using the fact that ϕ i is monotonic. Conversely, if y i ∈ ϕ i (y) for every i ∈ N, then r( y) = sup i∈N y i ∈ i∈N ϕ i (y) = ↑ y.
This remark makes it easier for us to show that r −1 (↑ y) is compact for every y ∈ Y . For each i 0 ∈ N, let Q i 0 = { y ∈ X | ∀i ≤ i 0 · y i ∈ ϕ i (y)}. Let K i 0 be the set of all elements y of Q i 0 such that y i = y i 0 for every i ≥ i 0 . Note that K i 0 is finite, (recall that each y i with i ≤ i 0 is taken from the finite set j≤i E j ), and that Q i 0 = ↑ K i 0 . Indeed, for every y ∈ Q i 0 , its image f i 0 ( y) by the idempotent deflation f i 0 is in K i 0 , and is below y. So Q i 0 is (finitary) compact. Every ωB-domain is stably compact [AJ94, Theorem 4.2.18], and any intersection of saturated compacts in a stably compact space is compact, so r −1 (↑ y) Let us now show that ↓ r(F ) is closed for every closed subset F of X. Consider a directed family (z j ) j∈J of elements of ↓ r(F ), and let z = sup j∈J z j . Since z j ∈ ↓ r(F ), F intersects r −1 (↑ z j ). The family (r −1 (↑ z j )) j∈J is a filtered family of compact saturated subsets of X, each of which intersects the closed set F . Since X is an ωB-domain, it is stably compact, hence well-filtered: so ↓ j∈J r −1 (↑ z j ) intersects F . (Explicitly: if it did not, it would be included in the open complement U of F , hence some r −1 (↑ z j ) would be included in U , contradicting the fact that it intersects F .) Let y be any element of ↓ j∈J r −1 (↑ z j ) ∩ F . Then z j ≤ r( y) for every j ∈ J, so z = sup j∈J z j ≤ r( y), hence z ∈ ↓ r(F ).

Products, Bilimits
We first show that finite products of QRB-domains are again QRB-domains.
Recall that a retraction p : X → Y , with section e : Y → X, is a projection iff, additionally, e(p(x)) ≤ x for every x ∈ X; then e is usually called an embedding, and is determined uniquely from p. An expanding system of dcpos is a family (X i ) i∈I , where I is a directed poset (with ordering ≤), with projection maps (p ij ) i,j∈I,i≤j where p ij : X j → X i , p ii = id X i , and p ik = p ij • p jk whenever i ≤ j ≤ k [AJ94, Section 3.3.2]. This is nothing else than a projective system of dcpos, where the connecting maps p ij must be projections. If e ij : X i → X j is the associated embedding, then one checks that e ii = id X i and e ik = e jk •e ij whenever i ≤ j ≤ k, so that (X i ) i∈I together with (e ij ) i,j∈I,i≤j forms an inductive system of dcpos as well. In the category of dcpos, the projective limit of the former coincides with the inductive limit of the latter (up to natural isomorphism), and is called the bilimit of the expanding system of dcpos. We write this bilimit as lim i∈I X i , leaving the dependence on ≤, p ij , e ij , implicit. This can be built as the dcpo of all those elements x = (x i ) i∈I ∈ i∈I X i such that p ij (x j ) = x i for all i, j ∈ I with i ≤ j, with the componentwise ordering.
General bilimits of countably-based dcpos will fail to be countably-based in general, so we shall restrict to bilimits of expanding sequences of dcpos [AJ94, Definition 3.3.6]: these are expanding systems of dcpos where the index poset I is N, with its usual ordering. To make it clear what we are referring to, we shall call ω-bilimit of spaces any bilimit of an expanding sequence (not system) of spaces. Bilimits are harder to deal with than products. But the difficulty was solved by Jung [Jun88, Section 4.1] in the case of RB-domains and deflations, and we proceed in a very similar way. We first recapitulate the notion of bilimit.
Consider any set G of functions ψ from X to Fin(X) such that ψ(x) ⊇ ↑ x, i.e., x ∈ ψ(x), for every x ∈ X. We say that G is qfs (for quasi-finitely separating) iff given any finitely Proposition 5.3. Let X be a poset. Then X is a QRB-domain iff X is a quasi-continuous dcpo and the set G of quasi-deflations on X is qfs.
Proof. If X is a QRB-domain, then let (↑ E k , x k ) ∈ Fin(X) × X be such that ↑ E k Î x k for every k, 1 ≤ k ≤ n, and (ϕ i ) i∈I be a generating family of quasi-deflations. For each k, And we may pick the same i for every k, by directedness. So ϕ i is the desired ψ ∈ G.
Let us show that H is directed. Pick ϕ and ϕ ′ from G.
Proof. Let (X i ) i∈I be an expanding system of QRB-domains, with projections p ij : X j → X i and embeddings e ij : X i → X j , i ≤ j. Let X = lim i∈I X i . There is a projection , and an embedding e i : X i → X for every i ∈ I.
We observe that: for every j ≥ i. Indeed, consider any directed family (y k ) k∈K such that p j ( x) ≤ sup k∈K y k . Then p i ( x) = p ij (p j ( x)) ≤ sup k∈K p ij (y k ), so for some k ∈ K, there is a z ∈ E with z ≤ p ij (y k ). Then e ij (z) ≤ e ij (p ij (y k )) ≤ y k . We conclude since e ij (z) ∈ Qe ij (↑ E).
We now claim that the family D x of all finitary compacts of the form Replacing i by k, ↑ E by the finitary compact Qe ik (↑ E), j by k, and ↑ E ′ by Qe jk (↑ E ′ ) if necessary, we can therefore simply assume that Moreover, we claim that In particular, X is a quasi-continuous dcpo.
We check that the set of quasi-deflations on X is qfs. Consider a finite collection of pairs (↑ D k , x k ) ∈ Fin(X)×X with ↑ D k Î x k , 1 ≤ k ≤ n. Recall that ↑ D k Î x k can be rephrased equivalently as: Proposition 3.5, for each k, pick Qe i (↑ E k ) ∈ D x k included in ↑ ↑ D k , in particular above ↑ D k . I.e., pick i ∈ I and ↑ E k ∈ Fin(X i ) such that ↑ E k Î p i ( x k ), and such that ↑ D k ⊇ Qe i (↑ E k ).
(We can pick the same i for every k, by directedness, as above.) Since X i is a QRB-domain, and ↑ E k Î p i ( x k ), using Proposition 3.5, there is a quasi-deflation ϕ on X i such that Consider ψ : X → Fin(X) defined as Qe i •ϕ•p i . Qe i , restricted to Fin(X i ), takes its values in Fin(X), using Lemma 3.7 and the fact that Qe i = (η X • e i ) † . Moreover, ψ is continuous from X to Fin V (X), hence to Fin σ (X) since X is quasi-continuous, by Corollary 3.6. For every x ∈ X, p i ( x) ∈ ϕ(p i ( x)), since ϕ is a quasi-deflation. Then e i (p i ( x)) is below x, and is in ψ( x), so x ∈ ψ( x). So ψ is a quasi-deflation.
Moreover, by construction, for each k, . So the set of quasi-deflations on X is qfs.
By Proposition 5.3, X is then a QRB-domain.
To deal with ω-bilimits of ωQRB-domains, observe that any bilimit of a countable expanding system (in particular, an expanding sequence) of countably-based quasi-continuous dcpos is countably-based. Indeed, a countably based quasi-continuous dcpo X i has a countable base of sets of the form The D x construction above, suitably modified, shows that the sets necessarily countable, base of the topology on X. By Proposition 3.18, X is an ωQRBdomain.

The Probabilistic Powerdomain
Let X be a fixed topological space, and let O(X) be the lattice of open subsets of X.
We write δ x for the Dirac valuation at x, a.k.a., the point mass at x. This is the continuous valuation such that δ x (U ) = 1 if x ∈ U , δ x (U ) = 0 otherwise.
The probabilistic powerdomain construction V 1 is an elusive one, and natural intuitions are often wrong. For example, one might imagine that if X has all binary least upper bounds, then so has V 1 (X). This was dispelled by Jones and Plotkin [JP89]. Consider X = {⊥, a, b, ⊤}, with a and b incomparable, ⊥ below every element and ⊤ above every element (see Figure 1, right). Then the upper bounds of 1 2 δ ⊥ + 1 2 δ a and 1 So there is no unique least upper bound; in fact, there are uncountably many of them, even on this small example.
It is unknown whether V 1 (X), with X = {⊥, a, b, ⊤} is an RB-domain, although it is an FS-domain, as a consequence of [JT98,Theorem 17]. Again, some of the most natural ideas one can have about V 1 (X) are flawed. It seems obvious indeed that V 1 (X) should be the bilimit of the sequence of finite posets V 1 n 1 (X), defined as those probability valuations (1 − α a − α b − α ⊤ )δ ⊥ + α a δ a + α b δ b + α ⊤ δ ⊤ where α a , α b , α ⊤ are integer multiples of 1 n . See Figure 4 for Hasse diagrams of a few of these posets, for n small.
That V 1 (X) is such a bilimit is necessarily wrong, because any bilimit of finite posets is an ωB-domain, hence is algebraic, but V 1 (X) is not algebraic, since no element except δ ⊥ is finite. However, one may imagine to define (non-idempotent) deflations f n on V 1 (X) directly, which would send ν ∈ V 1 (X) to some discretized probability valuation in V 1 n 1 (X). However, all known attempts fail. A careful study of [JT98] will make this precise. Let us only note that if we decide to define f n (ν) through its values on open sets, typically letting f n (ν)(U ) be the largest integer multiple of 1 n that is zero-or-strictly-below ν(U ), we obtain a set function that is not modular. If we decide to define f n ( x∈X α x δ x ) as x∈X β x δ x where for each x = ⊥ β x is the largest integer multiple of 1 n that is zero-or-strictly-below α x , then f n is not monotonic. If we decide to define f n (ν) as the largest probability valuation waybelow ν in V 1 n 1 (X), we run into the problem that there is no unique such largest probability valuation. For example, ν = 1 3 δ a + 1 3 δ b + 1 3 δ ⊤ admits four largest probability valuations in V 1 3 1 (X) way-below it: 1 3 δ ⊥ + 2 3 δ a , 1 3 δ ⊥ + 1 3 δ a + 1 3 δ b , 2 3 δ ⊥ + 1 3 δ ⊤ , and 1 3 δ ⊥ + 2 3 δ b , see Figure 5. Observe that the number of largest discretizations of ν in V 1 n 1 (X) is always finite, provided X is finite. This was our original intuition that replacing deflations by quasideflations, hence moving from RB-domains to QRB-domains, might provide a nice enough category of domains that would be stable under the probabilistic powerdomain functor V 1 . However, defining quasi-deflations directly, as hinted above, does not work either: monotonicity fails again. This is where the characterization of QRB-domains as quasiretracts of bifinite domains (up to details we have already mentioned) will be decisive.
If Y is a retract of X, then V 1 (Y ) is easily seen to be a retract of V 1 (X), using the V 1 endofunctor. We wish to show a similar result for quasi-retracts. We have not managed to do so. Instead we shall rely on the stronger assumptions that X is stably compact, that Y is a quasi-projection of X, not just a quasi-retract (i.e., the image of X under a proper map).
Moreover, we shall need to replace the Scott topology on V 1 (X) by the weak topology, which is the smallest one containing the subbasic opens [U > a], defined as {ν ∈ V 1 (X) | ν(U ) > a}, for each open subset U of X and a ∈ R. When X is a continuous pointed dcpo, the Kirch-Tix Theorem states that it coincides with the Scott topology (see [AMJK04], who attribute it to Tix [Tix95,Satz 4.10], who in turn attributes it to Kirch [Kir93, Satz 8.6]).
However, the weak topology is better behaved in the general case. For example, writing R + σ for R + ∪ {+∞} with the Scott topology, and [X → R + σ ] i for the space of all continuous maps from X to R + σ with the Isbell topology, there is a natural homeomorphism between the space of linear continuous maps from [X → R + σ ] i to R + σ and the space of of (extended, i.e., possibly taking the value +∞) continuous valuations on X, with the weak topology [Hec96,Theorem 8.1]. This is an analog of the Riesz Representation Theorem in measure theory, of which one can find variants in [Tix95,Gou07b] among others, and which we shall use silently in the proof of Theorem 6.5. Let V 1 wk (X) be V 1 (X) with its weak topology.
V 1 wk defines an endofunctor on the category of topological spaces, by V 1 wk (f )(ν)(V ) = ν(f −1 (V )), where f : X → Y , ν ∈ V 1 wk (X), and V ∈ O(Y ). That V 1 wk (f ) is continuous for every continuous f , in particular, is obvious, since for every open subset V of Y , As we have said above, we shall also require X to be stably compact. If this is so, then the cocompact topology on X consists of all complements of compact saturated subsets. Write X d , the de Groot dual of X, for X with its cocompact topology. Then X d is again stably compact, and X dd = X (see [AMJK04,Corollary 12] or [GHK + 03, Corollary VI-6.19]). The patch topology on X, mentioned earlier, is nothing else than the join of the two topologies of X and X d .
Write X patch for X equipped with its patch topology. If X is stably compact, then X patch is not only compact Hausdorff, but the graph of the specialization preorder ≤ of X is closed in X patch : one says that (X patch , ≤) is a compact pospace. The study of compact pospaces originates in Nachbin's classic work [Nac65]. Conversely, given a compact pospace (Z, ), i.e., a compact space with a closed ordering on it, the upwards topology on Z consists of those open subsets of Z that are upward closed in . The space Z ↑ , obtained as Z with the upwards topology, is then stably compact. Moreover, the two constructions are inverse of each other. (See [GHK + 03, Section VI-6].) If X and Y are stably compact, then f : X → Y is proper if and only if f : X patch → Y patch is continuous, and monotonic with respect to the specialization orderings of X and Y [GHK + 03, Proposition VI.6.23], i.e., if and only if f is a morphism of compact pospaces. Now, the structure of the cocompact topology on V 1 wk (X), when X is stably compact, is as follows. For every continuous valuation ν on X, following Tix [Tix95], define ν † (Q) as inf U ∈O(X),U ⊇Q ν(U ), for every compact saturated subset Q of X. Define Q ≥ a as the set of probability valuations ν such that ν † (Q) ≥ a. The sets Q ≥ a are compact saturated in V 1 wk (X), and Proposition 6.8 of [Gou10] even states that they form a subbase of compact saturated subsets. This means that the complements of the sets of the form Q ≥ a , Q compact saturated in X, a ∈ R, form a base of the topology of V 1 wk (X) d . A similar claim was already stated in [Jun04, last lines].
Lemma 6.1. Let X, Y be stably compact spaces, and r be a proper surjective map from X to Y . Then V 1 wk (r)(ν) † (Q) = ν † (r −1 (Q)), for every compact saturated subset Q of Y .
Proof. We must show that inf V ⊇Q ν(r −1 (V )) = inf U ⊇r −1 (Q) ν(U ), where V ranges over opens in Y and U over opens in X.
For every open V containing Q, U = r −1 (V ) is an open subset of X containing the compact saturated subset Conversely, for every open U containing r −1 (Q), we shall build an open subset V containing Q such that r −1 (V ) ⊆ U . This will establish inf V ⊇Q ν(r −1 (V )) ≤ inf U ⊇r −1 (Q) ν(U ), hence the equality.
Let us establish surjectivity. One possible proof goes as follows. Let M 1 (Z) denote the space of all Radon probability measures on the space Z. If X is stably compact, then M 1 (X patch ) is compact in the vague topology, and forms a compact pospace with the stochastic ordering, where µ is below µ ′ if and only if µ(U ) ≤ µ ′ (U ) for every open subset U of X [AMJK04, Theorem 31]. By [AMJK04, Theorem 36], there is an isomorphism between V 1 wk (X) and M ↑ 1 (X patch ). Now assume a second stably compact space Y . For two measurable spaces A and B, and f : A → B measurable, let M(f ) map the Radon measure µ to its image measure, whose value on the Borel subset E of B is µ(f −1 (E)). A standard result [Bou69, 2.4, Lemma 1] states that for any two compact Hausdorff spaces A and B, if r is continuous surjective from A to B, then M(r) is surjective. The desired result follows, up to a few technical details, by taking A = X patch , B = Y patch , remembering that since r is proper from X to Y , it is continuous from X patch to Y patch .
Instead of working out the-technically subtle but boring-technical details, let us give a direct proof, similar to the above cited Lemma 1, 2.4 [Bou69]. Instead of using the Hahn-Banach Theorem, we rest on the following Keimel Sandwich Theorem [Kei06, Theorem 8.2]: let C be a topological cone, q : C → R + σ be a continuous superlinear map, p : C → R + σ be a sublinear map, and assume q ≤ p; then there is a continuous linear map Λ : C → R + σ such that q ≤ Λ ≤ p. Here, a cone is an additive commutative monoid, with a scalar multiplication by elements of R + satisfying a(x + y) = ax + ay, (a + b)x = ax + bx, (ab)x = a(bx), 1x = x, 0x = 0 for all a, b ∈ R + , x, y ∈ C. A cone is topological if and only if addition and multiplication are continuous. The continuous maps f : C → R + σ are sometimes called lower semi-continuous in the literature. Such a map is superlinear (resp., sublinear, linear ) if and only if f (ax) = af (x) for all a ∈ R + , x ∈ C and f (x + y) ≥ f (x) + f (y) for all x, y ∈ C (resp., ≤, =). It is easy to see that the space [X → R + σ ] of all continuous maps from X to R + σ , equipped with the obvious addition and scalar multiplication and with the Scott topology of the pointwise ordering, is a topological cone. Putting together Proposition 6.3 and Proposition 6.4, we obtain: Theorem 6.5 (Key Claim). Let X be a stably compact space, and Y be a T 0 space. If r is a proper surjective map from X to Y , then V 1 wk (r) is a proper surjective map from V 1 wk (X) to V 1 wk (X).
In particular, if Y is a quasi-projection of X, then V 1 wk (Y ) is a quasi-projection of V 1 wk (X).
We shall apply this theorem twice, and first, to finite pointed posets. Let < be the strict part of ≤.
Definition 6.6 (Path Space). Let Y be any finite pointed poset. Write y → y ′ iff y is immediately below y ′ , i.e., y < y ′ and there is no z ∈ Y such that y < z < y ′ . A path π in Y is any set {y 0 , y 1 , . . . , y n } ⊆ Y with y 0 = ⊥ → y 1 → . . . → y n . The path space Π(Y ) is the set of paths in Y , ordered by ⊆.
Alternatively, the ordering on paths y 0 → y 1 → . . . → y n is the prefix ordering on sequences y 0 y 1 . . . y n .
Note that Π(Y ) is always a finite tree, i.e., a finite pointed poset such that the downward closure of a point is always totally ordered. Up to questions of finiteness, this is exactly how we built a tree from an ordering in the proof of Lemma 4.12, by the way.
We observe that every finite pointed poset Y is a quasi-projection of its path space Π(Y ).
Lemma 6.7. For every finite pointed poset Y , the map r : Π(Y ) → Y defined by r(π) = max π is proper and surjective.
Proof. See Figure 6, which displays the path space of the space Y of Figure 2 (i). Each gray region is labeled with an element from Y , which is the image by r of every point in the region; e.g., the top right, 5-element region is mapped to j in Y .
Formally, let X = Π(Y ), and define r : X → Y by r(π) = max π, i.e., r(y 0 → y 1 → . . . → y n ) = y n . The map r is surjective, and monotonic. Since X and Y are finite, r is then trivially proper.
Y is certainly not a retract of Π(Y ) in general: it is, if and only if Y is a tree. Indeed, if Y is a tree, then Y is isomorphic to Π(Y ), and conversely, every retract of a tree is a tree.
Finite trees are very special. Jung and Tix proved that V ≤1 (T ) is an RB-domain [JT98, Theorem 13] for every finite tree T . They noted (comment after op.cit.) that V ≤1 (T ) is even a bc-domain in this case, i.e., a pointed continuous dcpo in which every pair of elements with an upper bound has a least upper bound. It is well-known that every bc-domain is an RB-domain: given any finite subset A of a basis B of a bc-domain X, the map f A (x) = sup(A ∩ ↓ ↓ x) is a deflation, the family of these deflations is directed, and their least upper bound is the identity map.
Lemma 6.8. For every finite tree T , V 1 (T ) is a countably-based bc-domain.
Proof. Since T is a finite tree, it is trivially a continuous pointed dcpo, so V 1 (T ) is again continuous [Eda95, Section 3]. A basis is given by the valuations of the form t∈T a t δ t with a t ∈ [0, 1], t∈T a t = 1 and each a t rational. Since V 1 (T ) has a countable basis B, its topology has a countable base consisting of the subsets ↑ ↑ b, b ∈ B. So V 1 (T ) is countably-based.
To show that V 1 (T ) is a bc-domain, we observe that every probability valuation ν on T is entirely characterized by the values ν(↑ t), t ∈ T . Indeed, for every open subset U of T , let Min U be the (finite) set of minimal elements of U ; the sets ↑ t, t ∈ Min U , are pairwise disjoint, so ν(U ) = t∈Min U ν(↑ t). The map f : T → [0, 1] defined by f (t) = ν(↑ t) satisfies f (⊥) = 1 and f (t) ≥ t ′ ∈T,t→t ′ f (t ′ ) for every t ∈ T . Let us call such maps admissible.
Given any admissible map f , there is a unique probability valuation ν such that f (t) = ν(↑ t) for every t ∈ T , namely t∈T a t δ t with a t = f (t) − t ′ ∈T,t→t ′ f (t ′ ). So V 1 (T ) is orderisomorphic to the poset of admissible maps, with the pointwise ordering. Therefore we only have to show that any two admissible maps f 1 , f 2 below a third one f 0 have a least upper bound f . As a least upper bound, f (t) must be above f 1 (t), f 2 (t), and t ′ ∈T,t→t ′ f (t ′ ), so define f (t) by descending induction on t by f (t) = max(f 1 (t), f 2 (t), t ′ ∈T,t→t ′ f (t ′ )). (By descending induction, we mean induction on the largest length n of a sequence t 0 → t 1 → . . . → t n in T such that t 0 = t.) This is admissible if and only if f (⊥) = 1, and in this case will be the least upper bound of f 1 , f 2 . By definition f (⊥) ≥ 1. It is easy to see that f (t) ≤ f 0 (t) for every t, by descending induction on t: so f (⊥) ≤ f 0 (⊥) = 1, hence f is admissible.
We retrieve the Jung-Tix result that V ≤1 (T ) is a bc-domain for every tree T : let T ⊥ be T with an extra bottom element added below all elements of T , and apply Lemma 6.8 to V 1 (T ⊥ ) ∼ = V ≤1 (T ). Proposition 6.9. For every finite pointed poset Y , V 1 (Y ) is a continuous ωQRB-domain.
We can finally prove the main theorem of this paper.
Theorem 6.10. The probabilistic powerdomain of any ωQRB-domain is an ωQRB-domain.
Proof. Let Y be an ωQRB-domain. By Theorem 4.13, Y is the image of some ωB-domain X = lim i∈N X i under some proper surjective map. Since V 1 is a locally continuous functor on the category of dcpos, (as mentioned in proof of [JT98, Lemma 11]), V 1 (X) is also a bilimit of the spaces V 1 (X i ), i ∈ I. Each V 1 (X i ) is a continuous ωQRB-domain by Since X is bifinite, it is stably compact, (use, e.g., Theorem 3.14), and V 1 (X) = V 1 wk (X) because X is continuous and pointed, using the Kirch-Tix Theorem. So V 1 wk (Y ) is the image of V 1 (X) under a proper surjective map, by Theorem 6.5. It is clear that V 1 wk (Y ) is T 0 , so by Proposition 4.10 V 1 wk (Y )) is an ωQRB-domain in its specialization preorder , and its topology must be the Scott topology of .
Using the fact that V 1 (X) is continuous whenever X is continuous and pointed [Eda95, Section 3], it also follows: Corollary 6.11. The probabilistic powerdomain of any continuous ωQRB-domain (in particular, every RB-domain) is again a continuous ωQRB-domain.

Conclusion, Failures and Perspectives
We have shown that the category ωQRB of ωQRB-domains and continuous maps is a category of quasi-continuous, stably compact dcpos that is closed, not only under finite products, bilimits of expanding sequences, retracts (and even quasi-retracts), but also under the probabilistic powerdomain functor V 1 . It is thus reasonably well-behaved.
However, [T → T ] is not an ωQRB-domain. Assume (ϕ i ) i∈N were a generating sequence of quasi-deflations on [T → T ]. For each function f : N → {0, 1}, there is a continuous mapf : T → T that sends ⊥ to ⊥, ⊤ to ↓ i∈I ↑ f i (x) = ↑ x, while the converse inclusion is obvious. So (ϕ i ) i∈I is a generating family of quasi-deflations.
Defining the controlled ωQRB-domains as the ωQRB-domains, except with sequences of controlled quasi-deflations instead of directed families, and similarly for the ωFS-domains (a.k.a., the countably-based FS-domains, again a Cartesian-closed category [Jun90, Theorem 11]), we prove similarly: Theorem 7.4. The controlled ωQRB-domains are exactly the ωFS-domains, and hence form a Cartesian-closed category.
Using this last observation, Corollary 6.11 settles half of the conjecture that the probabilistic powerdomain of an ωFS-domain would be an ωFS-domain again. We are only lacking control.

Open Problems
(1) Is countability necessary in Theorem 4.13? Precisely, can one show that the QRBdomains are exactly the quasi-retracts of B-domains? The main difficulty seems to lie in the fact that a non-countable analog of Lemma 4.12 is missing-and Rudin's Lemma does not quite give us what we need, as discussed before the statement of the lemma.
(2) If Y is a quasi-retract of X, X is stably compact, and Y is T 0 , then is V 1 wk (Y ) a quasi-retract of V 1 wk (X)? This would be the analog of Theorem 6.5, only with quasiretractions instead of quasi-projections. (3) Is stable compactness necessary to derive Theorem 6.5? (4) One way of trying to prove that the probabilistic powerdomain of an ωFS-domain is again an ωFS-domain would be by inventing a new notion, say of good maps, and show that the ωFS-domains, or alternatively the controlled ωQRB-domains, are exactly the images under good maps of ωB-domains. Good maps should intuitively be intermediate between projections and proper surjective maps, in the sense that every projection should be good, and that every good map should be proper and surjective. Indeed surjective proper maps preserve the QRB part, but not the control, while projections preserve too much, in the sense that not all ωQRB-domains, only the ωRB-domains, are retracts of ωB-domains. Such a characterization of ωFS-domains would be of independent interest, too.