Domain theorists have learned to love algebraic and continuous dcpos. These notions have a relaxation, quasi-algebraic and quasi-continuous dcpos, that would deserve to be better known.
I have in fact worked on such domains, and I would like to tell about a nifty theorem that was independently found by Jimmie Lawson and Xiaoyong Xi [1], and by Achim Jung and myself [2] last year. This will have to wait until next time. For now, I would like to say a few words about the basics of quasi-continuous dcpos and locally finitary compact spaces, and the connection to so-called hypercontinuous lattices.
I must confess I’ve shied away from giving these objects the place they deserve in the book. Instead of being mentioned in definitions and theorems, they mostly occur in exercises. That was probably a mistake.
On the other hand, many properties of quasi-continuous domains are proved almost in the same way as for continuous domains. The most notable difference is a regular reliance on Rudin’s Lemma (Proposition 5.2.25).
Quasi-continuous posets
Quasi-continuous posets are defined and first studied in Exercise 5.1.34. The main difference with continuous posets is that you approximate points x not by single points y way-below x, but by finite sets E that are collectively below x. This is written E ≪ y, and means that every directed family (xi)i in I whose sup is above y contains an element xi that is above some element of E.
Although that looks like saying that E should contain a point way-below y, it is not so, because you may need to select different points from E, depending on the directed family (xi)i in I.
A typical example is given by the dcpo N2 of Figure 5.1, obtained by putting two copies of N side by side and adding an fresh element ω on top. For E, take any subset consist of one element m from the left copy of N and one element n from the right copy. Check that E ≪ ω, but neither m nor n is way-below ω. It all depends whether you reach ω from the left copy or from the right copy! Indeed, N2 is a quasi-continuous dcpo that is not continuous.
Locally finitary compact spaces
Locally finitary compact spaces are defined and first studied in Exercise 5.1.42. Recall that a space is locally compact if and only if, given any point x and any open neighborhood U of x, you can find a compact saturated subset Q inside U whose interior contains x. If we require that Q can be chosen finitary compact, that is, of the form ↑E with E finite, then we obtain the so-called locally finitary compact spaces.
We then learn (Exercise 5.2.31) that every quasi-continuous dcpo is locally finitary compact, and that, in fact, the quasi-continuous dcpos are exactly those dcpos which are locally finitary compact in their Scott topology.
Much later, in Exercise 8.2.15, we learn that every quasi-continuous dcpo is sober in its Scott topology, extending the well-known fact that every continuous dcpo is sober.
Finally, Exercise 8.3.39 asks you to show that the quasi-continuous dcpos are exactly the sober, locally finitary compact spaces (in particular, the topology of a sober, locally finitary compact space must be the Scott topology of its specialization ordering).
Other results are given in Exercise 9.1.21, Exercise 9.1.36, and Exercise 5.2.33.
A bit of history
The notion of locally finitary compact space originates with John Isbell [2], and occurs under a variety of names. Lawson and Xi call qc-spaces the T0 locally finitary compact spaces [1], for instance. The fundamental fact, mentioned above, that the sober locally finitary compact spaces are exactly the quasi-continuous dcpos is due to Bernard Banaschewski [4], as far as I know. This is also what the equivalence between Items (6) and (11) in Theorem 2 of [5] states.
In 1981, Gierz and Lawson showed that the Stone duals of locally finitary compact spaces where the hypercontinuous lattices. They have a very strange definition, and also many strange characterizations. I will spend the rest of this post explaining this. Among all possible definitions, I will select the one that I find clearest.
Hypercontinuous lattices
The question of the Stone duals of locally finitary compact spaces is to find the kind of lattice of open sets we obtain from those spaces.
Consider an arbitrary locally finitary compact space X. Since X is in particular locally compact, the frame L=O(X) of opens of X is a continuous frame, or equivalently a continuous, distributive complete lattice, by the Hofmann-Lawson Theorem (Theorem 8.3.21).
Here is the important new part. This is a bit tricky: you need to look at the Scott topology on L itself (not the topology of X).
Since L is continuous, the Scott topology has a basis of subsets of the form ↟U, where U is in L (an open subset of X). ↟U is the set of all V in L such that U is way-below V. Take any V in ↟U. By local finitary compactness, U is way-below V if and only if there is a compact saturated subset Q between U and V, and one can take Q finitary compact, that is, of the form ↑E for some finite E. The trick is standard: around each point x in Q, find a neighborhood of the form ↑Ex contained in U. Their interiors cover Q, so a finite union of them already cover Q, by compactness. Now take for E the corresponding finite union of sets Ex: ↑E contains Q, and is in particular U.
So, for every V in ↟U, we have found a finite set E such that U ⊆ ↑E ⊆ V. For each one of the finitely many points xi in E (1≤i≤n), xi is in V, so V intersects the closed subset ↓xi. Write Ui for the complement of ↓xi in X, so V is not included in Ui. But V and Ui are points of L, and as such, V is not below Ui in L, for any i. This means that V is in the complement U of ↓{U1, …, Un} (downward closure is taken in L here, not X).
That complement, U, must also be included entirely inside ↟U, because the opens in U are exactly those which contain ↑E. The important point is that U is open in the upper topology of the inclusion ordering on L, not the Scott topology.
So we have proved that, around every point V of ↟U in L, there is an open neighborhood, U, for the upper topology, inside ↟U. It follows that ↟U is not only Scott open, but also open in the upper topology. From that, and the fact that the upper topology is always coarser than the Scott topology, on any poset, we come quickly to the conclusion that the Scott and the upper topologies coincide on L.
This is one of the possible definitions of a hypercontinuous poset, one that is not in [6], but I am pretty sure you can find this in [7]:
A poset is hypercontinuous if and only if it is continuous, and its Scott topology coincides with its upper topology.
What we have just shown is that the lattice of open sets O(X) of opens of a locally finitary compact space X is hypercontinuous.
Apart from such lattices of open sets, there are a few hypercontinuous, distributive complete lattices in nature. [0, 1], with its usual ordering, is one, since the Scott opens (a, 1] are the upper opens obtained as the complements of ↓a. I will show below that the poset product of any family of hypercontinuous, distributive complete lattices is again hypercontinuous (and a distributive complete lattice, too). In particular, [0, 1]n, or the Hilbert cube [0, 1]N are hypercontinuous distributive complete lattices..
But let us conclude on Stone duality. As far as I know, the following is not in [6], but I am pretty sure you can find in in [7]. (I’ve left my copy at the office, and I’m currently home.)
Theorem [7]. The adjunction O ⊣ pt restricts to an equivalence between the category of sober, locally finitary compact spaces (=quasi-continuous dcpos) and the opposite of the category of hypercontinuous distributive complete lattices and frame homomorphisms.
This completes the picture shown in Figure 8.3 in the book.
Proof. To finish the proof of the theorem, we have to show that given a hypercontinuous distributive complete lattice L, X=pt(L) is a sober locally finitary compact space. We already know that X is sober and locally compact, by the Hofmann-Lawson Theorem.
Consider an arbitrary point x of X, and an open neighborhood U of x in X. Since X is locally compact, there is a compact saturated subset Q of U whose interior contains x. The set ⧠Q of all the open subsets that contain Q is Scott-open in L=pt(X). It contains U, so it is an open neighborhood of U in L. Since L is hypercontinuous, this is an open neighborhood in the upper topology as well. So there are finitely many open subsets U1, …, Un such that U is in the complement of ↓{U1, …, Un}, and this complement is in ⧠Q. The first condition means that U is included in no Ui, so we can find a point xi in U that is not in Ui. Let E = {x1, …, xn}. In particular, ↑E is included in U. One checks easily that every open neighborhood of ↑E must be in the complement of ↓{U1, …, Un}, hence in ⧠Q. In other words, every open neighborhood of ↑E contains Q. By taking the intersection of all these neighborhoods, ↑E itself contains Q, and is therefore a neighborhood of x. It follows that X is locally finitary compact. ⧠
I said that every product of hypercontinuous distributive complete lattices is again a hypercontinuous distributive complete lattice. Indeed, such products correspond through Stone duality to coproducts of sober, locally finitary compact spaces, which are clearly sober, locally finitary compact spaces again.
Again by Stone duality, since every continuous domain is quasi-continuous, it follows that every dual of a continuous domain, namely, every completely distributive complete lattice, is hypercontinuous.
Next time, we shall explore those (sober) locally finitary compact spaces that are coherent and compact… and we shall see a nifty coincidence of notions.
— Jean Goubault-Larrecq (November 30th, 2014)
[1] Gaolin Li and Luoshan Xu. QFS-Domains and their Lawson Compactness. Order (2013) 30, pp. 233–248.
[2] Jean Goubault-Larrecq and Achim Jung. QRB, QFS, and the Probabilistic Powerdomain. Proceedings of the 30th Intl. Conf. on Mathematical Foundations of Programming Semantics, ENTCS, pp. 170-185, 2014.
[2] John R. Isbell. Meet-continuous lattices, Symposia Mathematica 16 (1975), pp. 41–54, convegno sulla Topologica Insiemsistica e Generale, INDAM, Roma, Marzo 1973.
[4] Bernhard Banaschewski. Essential extensions of T0-spaces. General Topology and Applications, 7, pp. 233–246, 1977.
[5] Jimmie D. Lawson. The versatile continuous order. In Michael G. Main, Austin Melton, Michael W. Mislove, and David A. Schmidt, editors, Proc. 3rd MFPS Workshop, pages 134–160. Springer Verlag LNCS 298, 1987.
[6] Gerhard Gierz and Jimmie D. Lawson. Generalized continuous and hypercontinuous lattices. Rocky Mountain Journal of Mathematics, 11(2), 1981.
[7] Gerhard Gierz, Karl-Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael Mislove, Dana S. Scott. Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications, vol. 93, Cambridge University Press (2003).