Posets determined by countably many core-compact subspaces

Posted on September 21, 2024 by jgl

Let me return to a topic that I have addressed a few times in recent posts: under which conditions does the Scott topology of the product of two posets coincide with the product topology of the Scott topologies? We have seen quite a few cases already, and this month we will see that this holds for ccω-posets, namely posets whose Scott topology is determined by the Scott topology of a countably ascending sequence of sub-d-posets, each of them being core-compact in its Scott topology. The construction is due to Lawson and Xu. The ccω-posets are exactly certain forms of colimits of posets in the category Posσ of posets and Scott-continuous maps, and the whole point is that not only are those colimits also colimits in Top, but those form colimits in Top over which products distribute (see the post of previous month). Read the full post.

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When do products distribute over colimits in Top?

Posted on August 20, 2024 by jgl

In a recent paper, Lawson and Xu give a new class of posets on which the Scott topology of the (poset) product of two posets coincides with the product topology (of each poset with its Scott topology). I will explain that more in detail next month. The following question is at the heart of their approach, and this is what I will focus on this month. Given two diagrams F and G in Top, we can form their colimits colim F and colim G. We can also consider the product diagram F × G. Is it true that colim (F × G) = colim F × colim G? We will see that this is rarely the case—I will give counterexamples—and that one of the most general situations where that is actually true is given by what I will call ω-rigid diagrams of core-compact spaces. Read the full post.

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Happy summer holidays 2024!

Posted on July 16, 2024 by jgl

I will not post anything this month, sorry: I am on holidays, starting in a few days, and for about one month. Said otherwise: I have tried finding some time to write up on something before leaving off, but I will not manage to do so this time. Hence there will be no serious July 20th, 2024 post, sorry. I will definitely try to post something on August 20th, and I already have something in mind.

I am taking the opportunity to say that, after more than 10 years of posting every month, I may start slowing down a bit. Each post takes me between three days and one week of work, and it is getting harder and harder to find the time to write them.

In the meantime, you might want to attend the 10th International Symposium on Domain Theory, which will take place in Qufu, Shandong, China, 2-5 August 2024. I won’t be there (remember, I will be on holidays), but many others will. Otherwise, let me wish you a happy summer holiday!

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The complete lattice Lfan (part II)

Posted on June 20, 2024 by jgl

Last time, we had started to study the complete lattice Lfan, namely just N × N, with equality as ordering on the first component and the usual ordering on the second component, plus an extra bottom element ⊥, plus an extra top element ⊤. This is a complete lattice with the Chen-Kou-Lyu property, and we have seen a few consequences of that. This time, we will see that Lfan is a maximal limit space (in particular, it is sober and weakly Hausdorff); we knew that it is not core-compact, but it is consonant; we knew that it is not first-countable, but its lattice of open sets is second-countable in its Scott topology; and the upper Vietoris and Scott topologies coincide on its Smyth hyperspace. Read the full post.

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On products of dcpos, the Miao-Xi-Li-Zhao lemma, and the complete lattice Lfan (part I)

Posted on May 20, 2024 by jgl

The product of a poset P with itself can be given two topologies: the Scott topology of the product, or the product of the Scott topologies. Those two topologies differ in general, but they coincide when P is a continuous poset, and more generally when P is core-compact, or first-countable. We will see a (very) simple example of a dcpo—even a complete lattice—for which those two topologies coincide, but that is neither core-compact nor first-countable. This was discovered independently by Xu, Shen, Xi and Zhao in 2020, by Chen, Kou and Lyu in 2022 and by Hertling in 2022. Showing that the Scott topology of the product coincides with the product of the Scott topologies is best done through a lemma due to Miao, Xi, Li, and Zhao in 2022, and which says the curious thing that this coincidence of topologies happens as soon as there are only countably many ideals that have suprema on the given poset. Read the full post.

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Hoover’s maximal limit spaces II: products, liftings, retracts, function spaces, and hyperspaces

Posted on April 20, 2024 by jgl

Last time, we had introduced Hoover’s maximal limit spaces: spaces in which every convergent filter has a unique largest limit. That notion is closed under many constructions, as we will see: products, liftings, retracts, notably, and that is elementary. The space of continuous functions from a core-compact space to a maximal limit space is also a maximal limit space, as shown by Hoover. That is a more difficult theorem, which will rely on the characterizations we saw last time of maximal limit spaces as weakly Hausdorff spaces with some additional properties. We will also look at the Smyth and Hoare hyperspaces. Read the full post.

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Hoover’s maximal limit spaces I: local strong sobriety, bounded sup-completeness and weak Hausdorffness

Posted on March 20, 2024 by jgl

In 1995, Douglas Hoover introduced and studied a notion of maximal limit spaces: spaces in which every convergent net has a unique largest limit. This has connections with many other kinds of spaces that we have explored already, and I will show you one of Hoover’s theorems, suitably rephrased in light of a previous post of mine on weakly Hausdorff spaces (in the sense of Keimel and Lawson): the maximal limit spaces are exactly the locally strongly sober, bounded-sup-complete spaces, and therefore also the weakly Hausdorff, coherent, sober, bounded-sup-complete spaces. Read the full post.

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Skula spaces II: the Dow-Watson counterexample

Posted on February 20, 2024 by jgl

Last time, we have started to explain some results due to A. Dow and S. Watson, and we have seen that every compact T0 scattered space of scattering height at most 3 is Skula, namely can be obtained from a (necessarily Noetherian) space X by giving it its Skula topology instead. Today I will explain an example, also due to Dow and Watson, of a compact T0 scattered space of scattering height 4 that is not Skula. Read the full post.

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Skula spaces I: clopen selectors

Posted on January 21, 2024 by jgl

A Skula space is a space that is obtained from another space X by giving it the Skula topology instead, which is generated by the open sets and the closed sets of the original space X. In 1990, Alan Dow and Stephen Watson gave a simple necessary and sufficient condition for a compact Hausdorff space to be Skula. I will show what it is—it involves the notion of clopen selectors in the title—, how it relates to the theory of Noetherian spaces, and I will end with a corollary of theirs about compact Hausdorff scattered spaces of scattering height at most 3 being Skula. That will be an opportunity to talk about scattered spaces, totally disconnected spaces, connected and quasi-connected components. I would also like start the full post with a tribute to Maurice Pouzet, a nice human fellow and a great mathematician, who passed away on December 31st, 2023.

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The (positive) unique structure theorem

Posted on December 21, 2023 by jgl

There is an amazing theorem in the theory of analytic spaces, called the unique structure theorem: any two countable families F1 and F2 of Borel subsets of an analytic space X that separate the points of X generate the same σ-algebra. F. Clerc, N. Fijalkow, B. Klin and P. Panangaden used a generalization of this in a 2019 paper about labelled Markov processes, which they call the positive unique structure theorem. It too, applies to any analytic space, and I will show that it even generalizes to a larger class of spaces, which I will call quasi-analytic; those are the continuous images of M. de Brecht’s quasi-Polish spaces (inside an arbitrary topological space, not necessarily Hausdorff or even T0). The treatment I will give is a mix of measure-theoretic and domain-theoretic techniques. Read the full post.

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