Zhao, Xi and Chen’s well-filtered, non-sober dcpo

There are several known examples of dcpos that are well-filtered, but not sober, and I have already mentioned one due to Xiaodong Jia. I would like to explain another one, due to Dongsheng Zhao, Xiaoyong Xi, and Yixiang Chen. This is a very simple modification of Johnstone’s non-sober dcpo J. Contrarily to Xiaodong Jia’s dcpo (and to J), it is uncountable, but it may be easier to see why it must be well-filtered: everything mostly boils down to a cardinality argument, or rather, as I will argue, to the properties of so-called regular ordinals. Read the full post.

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Quasi-Polish spaces as rounded ideal completions

This month, a pearl by Matthew de Brecht. It is known that the rounded ideal completion of an abstract basis (a set B with a transitive, interpolative relation) is a continuous dcpo, and that all continuous dcpos can be obtained this way. What do you get if you remove the requirement of interpolation? Well, and assuming B countable… exactly the quasi-Polish spaces! Read the full post.

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Convergence without points

Can you define convergence without mentioning points? More precisely, is there any form of Stone duality for convergence spaces, instead of just topological spaces? The short answer is yes. For the long answer, read the full post.

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X. Jia’s well-filtered, non-sober dcpo

[Business as usual, despite all viruses!] Peter Johnstone once showed the existence of a dcpo J that is not sober in its Scott topology. That dcpo is not well-filtered either. Is there a dcpo that is not sober but is well-filtered? That is true, and the first one who found an example is Hui Kou. Since then, Xi and Zhao have also given another example, and I would like to describe another example of such a dcpo, due to Xiaodong Jia in his PhD thesis. Both Xi and Zhao’s example and Jia’s example are pretty simple spaces, but X. Jia’s example is countable. Read the full post.

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Dcpos built as graphs of functions

Let X and P be two dcpos, and let ψ be a map from X to P. When is the graph of ψ a dcpo? I will give you a funny sufficient condition, which involves the so-called d-topology, and Hausdorffness. I will briefly explain how this can be used to show that every Π02 subset of a continuous dcpo is domain-complete, namely, is homeomorphic to a Gδ subset of some other continuous dcpo. Read the full post.

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Borel sets, analytic sets, and the Baire property

Let me venture into the realm of σ-algebras. Yes, you might say, that is measure theory, not topology… but topology plays an important role in measure theory and, for that matter, descriptive set theory. I will tell you about sets with the Baire property. Those are pretty simple objects, or at least they appear to be simpler than Borel sets, but we will see that this is the other way around: all Borel sets have the property of Baire. The proof is pretty easy, as well. I will also spend some more time to explain a more complicated result, due to O.M. Nikodým, and which says that all A-sets have the property of Baire as well, in a second-countable space. None of that ever uses any Hausdorffness, or in fact any separation property whatsoever. Read the full post.

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The Rasiowa-Sikorski lemma and the Baire property

Merry Christmas! And a Happy New Year, too. There are incredible links between logic and topology, and I would like to start with something called the Rasiowa-Sikorski lemma in logic. That is a theorem that states the existence of certain prime filters in a Boolean algebra, and which was invented as a clean justification of a completeness argument for first-order logic. Robert Goldblatt realized in a 2012 paper that the Rasiowa-Sikorski lemma is a consequence of the fact that all compact Hausdorff spaces are Baire (plus some Stone duality, which is the main bridge between logic and topology). This is a beautiful argument… read the full post.

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Quotients, colimits of dcpos, and related matters

How do we build colimits in the category Top of topological spaces? This is easy: we take the quotient of a big disjoint sum. How do we build colimits in the category Dcpo of dcpos? This is a much more complicated question. All colimits indeed exist in Dcpo, and this has been shown by various authors over time, but this is complex. I will explain why. Then I will explain what the definition of quotients should be in Dcpo, and how one can build them. Funnily, this is related to other questions, such as the existence of d-completions, for example. Read the full post.

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Core-compact+well-filtered T0=sober locally compact

Last time, I motivated the construction of the well-filterification Wf(X) of a space X of X. Xu, Ch. Shen, X. Xi and D. Zhao by saying that it was needed to understand their proof of the fact that every core-compact well-filtered T0 space is sober, and hence also locally compact. This solves a question asked by X. Jia, and first solved positively by J. Lawson and X. Xi. I then realized that their proof contains a (somewhat concealed) gem; polishing it reveals an interesting new variant of the Hofmann-Mislove theorem, which applies to well-filtered spaces rather than to sober spaces, and rests on a funny countability assumption. We will see that the latter is satisfied in all core-compact spaces, and this will allow us to show that all core-compact well-filtered T0 spaces are locally compact, hence sober. Read the full post.

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Well-filterifications

Xiaodong Jia once asked the following question: is every core-compact, well-filtered space automatically locally compact? The question was solved positively this year by J. Lawson and X. Xi. I originally planned to try and explain their result. Even more recently, X. Xu, Ch. Shen, X. Xi and D. Zhao found a simpler solution, and I have changed my plans. My new plan for this time, and next time, is to explain what they have done. This time, we will concentrate on well-filterifications of topological space, which are just like sobrifications except ‘sober’ is replaced by ‘well-filtered T0‘. Building one is not completely obvious. Xu, Shen, Xi and Zhao show that the will-filterification of X can be defined as its set of closed WD sets, a new notion that is intermediate between directed sets and irreducible sets; the proof also relies on a refinement of R. Heckmann and K. Keimel’s topological version of Rudin’s Lemma, which is interesting in its own right. Read the full post.

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