{"id":521,"date":"2014-11-30T17:47:57","date_gmt":"2014-11-30T16:47:57","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=521"},"modified":"2022-11-19T15:31:31","modified_gmt":"2022-11-19T14:31:31","slug":"quasi-continuous-dcpos-and-locally-finitary-compact-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=521","title":{"rendered":"Quasi-continuous dcpos and locally finitary compact spaces"},"content":{"rendered":"

Domain theorists have learned to love algebraic and continuous dcpos.\u00a0 These notions have a relaxation, quasi-algebraic and quasi-continuous dcpos, that would deserve to be better known.<\/p>\n

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.\u00a0 This will have to wait until next time.\u00a0 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.<\/p>\n

I must confess I’ve shied away from giving these objects the place they deserve in the book<\/a>.\u00a0 Instead of being mentioned in definitions and theorems, they mostly occur in exercises.\u00a0 That was probably a mistake.<\/p>\n

On the other hand, many properties of quasi-continuous domains are proved almost in the same way as for continuous domains.\u00a0 The most notable difference is a regular reliance on Rudin’s Lemma (Proposition 5.2.25).<\/p>\n

Quasi-continuous posets<\/strong><\/p>\n

Quasi-continuous posets are defined and first studied in Exercise 5.1.34.\u00a0 The main difference with continuous posets is that you approximate points x<\/em> not by single points y<\/em> way-below\u00a0x, but by finite sets E<\/em> that are collectively<\/em> below x<\/em>.\u00a0 This is written E<\/em>\u00a0\u226a y<\/em>, and means that every directed family (xi<\/sub><\/em>)i<\/sub><\/em> in <\/sub>I<\/sub><\/em> whose sup is above y<\/em> contains an element xi<\/sub><\/em> that is above some element of E<\/em>.<\/p>\n

Although that looks like saying that E<\/em> should contain a point way-below y<\/em>, it is not so, because you may need to select different points from E<\/em>, depending on the directed family (xi<\/sub><\/em>)i<\/sub><\/em> in <\/sub>I<\/sub><\/em>.<\/p>\n

A typical example is given by the dcpo N<\/em>2<\/sub> of Figure 5.1, obtained by putting two copies of N<\/strong> side by side and adding an fresh element \u03c9 on top.\u00a0 For E<\/em>, take any subset consist of one element m<\/em> from the left copy of N<\/strong> and one element n<\/em> from the right copy.\u00a0 Check that E <\/em>\u226a \u03c9, but neither m<\/em> nor n<\/em> is way-below \u03c9.\u00a0 It all depends whether you reach \u03c9 from the left copy or from the right copy!\u00a0 Indeed, N<\/em>2<\/sub> is a quasi-continuous dcpo that is not continuous.<\/p>\n

Locally finitary compact spaces<\/strong><\/p>\n

Locally finitary compact spaces are defined and first studied in Exercise 5.1.42.\u00a0 Recall that a space is locally compact if and only if, given any point x<\/em> and any open neighborhood U<\/em> of x<\/em>, you can find a compact saturated subset Q<\/em> inside U<\/em> whose interior contains x<\/em>.\u00a0 If we require that Q<\/em> can be chosen finitary compact, that is, of the form \u2191E<\/em> with E<\/em> finite, then we obtain the so-called locally finitary compact spaces.<\/p>\n

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.<\/p>\n

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.<\/p>\n

Finally, Exercise 8.3.39 asks you to show that the quasi-continuous dcpos are exactly<\/em> 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).<\/p>\n

Other results are given in Exercise 9.1.21, Exercise 9.1.36, and Exercise 5.2.33.<\/p>\n

A bit of history<\/strong><\/p>\n

\n

The notion of locally finitary compact space originates with John Isbell [2], and occurs under a variety of names.\u00a0 Lawson and Xi call qc-spaces the T0<\/sub> locally finitary compact spaces [1], for instance.\u00a0 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.\u00a0 This is also what the equivalence between Items (6) and (11) in Theorem 2 of [5] states.<\/p>\n

In 1981, Gierz and Lawson showed that the Stone duals of locally finitary compact spaces where the hypercontinuous<\/em> lattices.\u00a0 They have a very strange definition, and also many strange characterizations.\u00a0 I will spend the rest of this post explaining this.\u00a0 Among all possible definitions, I will select the one that I find clearest.<\/p>\n

Hypercontinuous lattices<\/strong><\/p>\n

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.<\/p>\n

Consider an arbitrary locally finitary compact space X<\/em>.\u00a0 Since X<\/em> is in particular locally compact, the frame L<\/em>=O<\/strong>(X<\/em>) of opens of X<\/em> is a continuous frame, or equivalently a continuous, distributive complete lattice, by the Hofmann-Lawson Theorem (Theorem 8.3.21).<\/p>\n

Here is the important new part.\u00a0 This is a bit tricky: you need to look at the Scott topology on L<\/em> itself (not the topology of X<\/em>).<\/p>\n

Since L<\/em> is continuous, the Scott topology has a basis of subsets of the form \u219fU<\/em>, where U<\/em> is in L<\/em> (an open subset of X<\/em>).\u00a0 \u219fU<\/em> is the set of all V<\/em> in L<\/em> such that\u00a0U is way-below V<\/em>.\u00a0 Take any V<\/em> in \u219fU<\/em>.\u00a0 By local finitary compactness, U<\/em> is way-below V<\/em> if and only if there is a compact saturated subset Q <\/em>between U<\/em> and V<\/em>, and one can take Q<\/em> finitary compact, that is, of the form\u00a0\u2191E<\/em> for some finite E<\/em>.\u00a0 The trick is standard: around each point x<\/em> in Q<\/em>, find a neighborhood of the form \u2191Ex<\/sub><\/em> <\/em>contained in U.<\/em>\u00a0 Their interiors cover Q<\/em>, so a finite union of them already cover Q<\/em>, by compactness.\u00a0 Now take for E<\/em> the corresponding finite union of sets Ex<\/sub><\/em><\/em>: \u2191E<\/em> contains Q<\/em>, and is in particular U.<\/em><\/p>\n

So, for every V<\/em> in \u219fU<\/em>, we have found a finite set E<\/em> such that U \u2286 <\/em>\u2191E<\/em> \u2286<\/em> V<\/em>.\u00a0 For each one of the finitely many points xi<\/sub><\/em> in E<\/em> (1\u2264i<\/em>\u2264n<\/em>), xi<\/sub><\/em> is in V<\/em>, so V<\/em> intersects the closed subset \u2193xi<\/sub><\/em>.\u00a0 Write U<\/em>i<\/sub><\/em> for the complement of \u2193<\/em>xi<\/sub><\/em> in X<\/em>, so V<\/em> is not included in U<\/em>i<\/sub><\/em>.\u00a0 But V<\/em> and U<\/em>i<\/sub><\/em> are points of L<\/em>, and as such, V<\/em> is not below U<\/em>i<\/sub><\/em> in L<\/em>, for any i<\/em>.\u00a0 This means that V<\/em> is in the complement U<\/strong><\/em> of \u2193<\/em>{U<\/em>1<\/sub><\/em>, …, Un<\/sub><\/em>} (downward closure is taken in L<\/em> here, not X<\/em>).<\/p>\n

That complement, U<\/em><\/strong>, must also be included entirely inside \u219fU<\/em>, because the opens in U<\/em><\/strong> are exactly those which contain \u2191E<\/em>.\u00a0 The important point is that U<\/strong><\/em> is open in the upper<\/em> topology of the inclusion ordering on L<\/em>, not the Scott topology.<\/p>\n

So we have proved that, around every point V<\/em> of \u219fU<\/em> in L<\/em>, there is an open neighborhood,\u00a0U<\/em><\/strong>, for the upper<\/em> topology, inside \u219fU<\/em>.\u00a0 It follows that \u219fU<\/em> is not only Scott open, but also open in the upper topology.\u00a0 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<\/em>.<\/p>\n

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]:<\/p>\n

A poset is hypercontinuous if and only if it is continuous, and its Scott topology coincides with its upper topology.<\/p><\/blockquote>\n

What we have just shown is that the lattice of open sets O<\/strong>(X<\/em>) of opens of a locally finitary compact space X<\/em> is hypercontinuous.<\/p>\n

Apart from such lattices of open sets, there are a few hypercontinuous, distributive complete lattices in nature.\u00a0 [0, 1], with its usual ordering, is one, since the Scott opens (a<\/em>, 1] are the upper opens obtained as the complements of \u2193a<\/em>.\u00a0 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).\u00a0 In particular, [0, 1]n<\/sup><\/em>, or the Hilbert cube [0, 1]N<\/sup><\/strong> are hypercontinuous distributive complete lattices..<\/p>\n

But let us conclude on Stone duality.\u00a0 As far as I know, the following is not in [6], but I am pretty sure you can find in in [7].\u00a0 (I’ve left my copy at the office, and I’m currently home.)<\/p>\n

Theorem [7].<\/strong>\u00a0 The adjunction O<\/strong> \u22a3 pt<\/strong> 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.<\/p>\n

This completes the picture shown in Figure 8.3 in the book<\/a>.<\/p>\n

Proof.<\/em> To finish the proof of the theorem, we have to show that given a hypercontinuous distributive complete lattice L<\/em>, X=<\/em>pt<\/strong>(L<\/em>) is a sober locally finitary compact space.\u00a0 We already know that X<\/em> is sober and locally compact, by the Hofmann-Lawson Theorem.<\/p>\n

Consider an arbitrary point x<\/em> of X<\/em>, and an open neighborhood U<\/em> of x<\/em> in X<\/em>.\u00a0 Since X<\/em> is locally compact, there is a compact saturated subset Q<\/em> of U<\/em> whose interior contains x<\/em>.\u00a0 The set \u29e0Q<\/em> of all the open subsets that contain\u00a0Q<\/em> is Scott-open in\u00a0L<\/em>=pt<\/strong>(X<\/em>).\u00a0 It contains\u00a0U<\/em>, so it is an open neighborhood of\u00a0U<\/em> in\u00a0L<\/em>.\u00a0 Since\u00a0L<\/em> is hypercontinuous, this is an open neighborhood in the upper topology as well.\u00a0 So there are finitely many open subsets U<\/em>1<\/sub><\/em>, …, Un<\/sub><\/em> such that U<\/em> is in the complement of \u2193{U<\/em>1<\/sub><\/em>, …, Un<\/sub><\/em>}, and this complement is in \u29e0Q<\/em>.\u00a0 The first condition means that U<\/em> is included in no U<\/em>i<\/sub><\/em>, so we can find a point x<\/em>i<\/sub><\/em> in U<\/em> that is not in U<\/em>i<\/sub><\/em>.\u00a0 Let E<\/em> = {x<\/em>1<\/sub><\/em>, …, xn<\/sub><\/em>}.\u00a0 In particular, \u2191E<\/em> is included in U<\/em>.\u00a0 One checks easily that every open neighborhood of \u2191E<\/em> must be in the complement of \u2193<\/em>{U<\/em>1<\/sub><\/em>, …, Un<\/sub><\/em>}, hence in \u29e0Q<\/em>.\u00a0 In other words, every open neighborhood of \u2191E<\/em> contains Q<\/em>.\u00a0 By taking the intersection of all these neighborhoods, \u2191E<\/em> itself contains Q<\/em>, and is therefore a neighborhood of x<\/em>.\u00a0 It follows that X<\/em> is locally finitary compact.\u00a0\u00a0\u00a0\u00a0 \u29e0<\/p>\n

I said that every product of hypercontinuous distributive complete lattices is again a hypercontinuous distributive complete lattice.\u00a0 Indeed, such products correspond through Stone duality to coproducts of sober, locally finitary compact spaces, which are clearly sober, locally finitary compact spaces again.<\/p>\n

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.<\/p>\n

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.<\/p>\n

\u2014 Jean Goubault-Larrecq<\/a>\u00a0(November 30th, 2014)\"jgl-2011\"<\/p>\n

[1] Gaolin Li and Luoshan Xu.\u00a0 QFS-Domains and their Lawson Compactness<\/a><\/em>. Order (2013) 30, pp. 233\u2013248.<\/p>\n

[2] Jean Goubault-Larrecq and Achim Jung. QRB, QFS, and the Probabilistic Powerdomain<\/a><\/em>.\u00a0 Proceedings of the 30th Intl. Conf. on Mathematical Foundations of Programming Semantics, ENTCS, pp. 170-185, 2014.<\/p>\n

[2] John R. Isbell. Meet-continuous lattices<\/em>, Symposia Mathematica 16 (1975), pp. 41\u201354, convegno sulla Topologica Insiemsistica e Generale, INDAM, Roma, Marzo 1973.<\/p>\n

[4] Bernhard Banaschewski. Essential extensions of T0<\/sub>-spaces<\/em>. General Topology and Applications, 7, pp. 233\u2013246, 1977.<\/p>\n

[5] Jimmie D. Lawson. The versatile continuous order<\/a><\/em>. In Michael G. Main, Austin Melton, Michael W. Mislove, and David A. Schmidt, editors, Proc. 3rd MFPS Workshop, pages 134\u2013160. Springer Verlag LNCS 298, 1987.<\/p>\n

[6] Gerhard Gierz and Jimmie D. Lawson.\u00a0 Generalized continuous and hypercontinuous lattices.<\/em>\u00a0 Rocky Mountain Journal of Mathematics, 11(2), 1981.<\/p>\n

[7] Gerhard Gierz, Karl-Heinrich Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael Mislove, Dana S. Scott.\u00a0 Continuous Lattices and Domains.<\/em>\u00a0 Encyclopedia of Mathematics and its Applications, vol. 93, Cambridge University Press (2003).<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Domain theorists have learned to love algebraic and continuous dcpos.\u00a0 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 … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"_crdt_document":"","footnotes":""},"class_list":["post-521","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/521","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=521"}],"version-history":[{"count":15,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/521\/revisions"}],"predecessor-version":[{"id":5963,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/521\/revisions\/5963"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=521"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}