Sober subspaces and the Skula topology

It often happens that one wishes to show that a certain subspace A of a given sober space X is sober. The following is a pearl due to Keimel and Lawson, which was mentioned to me by Zhenchao Lyu in July: the sober subspaces of a sober space are exactly its subsets that are closed in the Skula topology. Read the full post.

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Bc-hulls and Clat-hulls

Bounded-complete domains (bc-domains) are an incredibly useful form of continuous dcpos. Given a continuous dcpo X, is there a completion of X as a bc-domain, for example a free bc-domain on X? That does not exist in general, but Yuri Ershov showed that one can build a so-called bc-hull of any continuous dcpo in 1997. I will describe what that is in the full post. My point is really to show that, despite the fact that it is a complicated construction in general, this really becomes a very simple, and familiar, one when X is coherent.

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Shimrat’s theorem

Marcus Tressl just found a horrible mistake in the book: I had claimed that the T0 quotient of the topological quotient of any sober space by any equivalence relation is sober, but that is completely wrong. In fact, Moshe Shimrat had shown in 1956 that you can get absolutely any topological space as a topological quotient of a Hausdorff space. In the full post, I will explain why Shimrat’s theorem directly contradicts my claim, and I will explain Shimrat’s proof. I will also comment of my own personal (hence biased) selection of the papers I think were the best among those presented at the 8th International Symposium on Domain Theory, in Yangzhou, Jiangsu province, China, from which I have just come back.

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On countability: the compact completed sequence

Recently, Matthew de Brecht sent me a proof of a neat and rather surprising result: the product and the Scott topologies coincide for products of first-countable, not necessarily continuous, posets. This rests on a clever argument, inspired by techniques invented by Matthias Schröder, and a simple observation: if you take all the elements of a convergent sequence, plus one (any) of its limits, what you get is a compact set. The latter fails if you take a net instead of a sequence. Read the full post.

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Isbell’s non sober complete lattice

Johnstone space J (1981) is the most famous example of a non-sober dcpo. In 1982, Isbell came up with a non-sober complete lattice. His construction is so complex that most authors use it as a black box. I would like to explain how Isbell’s non-sober complete lattice is constructed. As you can expect, this is a pretty clever construction, but I claim this is understandable. I will conclude with remarkable results of Xu, Xi, and Zhao (2019), who proved that there even exists a non-sober frame. Read the full post.

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Countably presented locales

Reinhold Heckmann showed the following in a very nice paper of 2014: every countably presented locale is spatial. What makes it even nicer is that he shows how tightly this is connected with the Baire property. This also gives a localic description of Matthew de Brecht’s quasi-Polish spaces, and of Ruiyuan Chen’s more recent countably correlated spaces. Read the full post.

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A core-compact, non-locally compact space

Last time, I had announced that we would do Exercise V-5.25 of the red book, constructing a core-compact, yet not locally compact, space.  And this is exactly what we shall do: read the full post.

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Bernstein subsets of R

This month, we will start to do Exercise V-5.25 of the red book (Continuous Lattices and Domains), which gives an example of a core-compact, not locally compact space.  That is pretty hard to obtain, really.  This month, we will do the first step of that exercise by constructing Bernstein sets, which are very, very pathological subsets of R.  Read the full post.

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On countability

Let me first wish you a Merry Christmas, and since I will not post again next week, a Happy New Year 2019 as well.  I have no specific present this year, sorry… This month’s post is about a few thoughts I have had about the role of countability is a few situations other than the expected ones.  This will culminate in a recent result of de Brecht and Kawai—the Scott and the upper Vietoris topologies coincide on the Smyth powerdomain of a well-filtered second-countable space— and its clever proof: see the full post.

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The locale of random elements of a space

Alex Simpson has a lot of slides with very interesting ideas.  One of them is what he calls the locale of random sequences.  This is a terribly clever idea that aims at solving the question “what are random sequences?”, using locale theory.  He obtains a very big locale, but without points in general… because every single random sequence is essentially not random.  Read the full post.

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