The space S0

S0 is a space that occurs in Matthew de Brecht’s generalized Hurewicz theorem for quasi-Polish spaces, published in 2018. S0 is very simple: it is an infinite countably-branching tree, and if you order it so that the root is at the top, S0 comes with the upper topology of the resulting ordering. S0 is one of the four canonical examples of a non-quasi-Polish space (in a precise sense). I will describe it, and I will show how closed sets and compact saturated sets in S0 can be described through certain kinds of subtrees. With that done, we will see that S0 is sober, Choquet-complete, and completely Baire, but not locally compact, not convergence Choquet-complete, not compactly Choquet-complete, not LCS-complete, and, finally, not quasi-Polish. Read the full post.

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Aliaume Lopez’ master theorem of Noetherian spaces

There are quite a few constructions that we can use to build new Noetherian spaces from old ones: spaces of finite words, of finite trees (as in Section 9.7 of the book), and a few others. Instead of writing a new proof each time, is there some form of master theorem that would have all those results as corollaries? This is exactly what Aliaume Lopez found this year. Read the full post.

Oh, and Season’s greetings, too!

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Weakly Hausdorff spaces, and locally strongly sober spaces

A funny convergence of topics happened a few weeks ago. Frédéric Mynard told me about so-called locally strongly sober spaces (which, I am ashamed to say, I had heard about but completely forgotten about). At the same time, I was interested in so-called weakly Hausdorff spaces, as defined by Klaus Keimel and Jimmie Lawson in their paper on measure extension theorems for T0 spaces. I realized that those two classes of spaces had a lot in common, and this led me to inquire whether that was a coincidence. As you may guess, this is not: we will see that the locally strongly sober spaces are exactly the weakly Hausdorff, coherent sober spaces. Read the full post.

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Strongly compact sets and the double hyperspace construction

The notion of strongly compact set is due to Reinhold Heckmann. A few months ago, I said that I would explain why the sobrification of the space Qfin(X) of finitary compact sets on a sober space X is not the Smyth hyperspace Q(X), rather its subspace of strongly compact saturated sets Qs(X). This what I will start with. I will then present a funny other case where strongly compact sets are required. There is a long line of research purporting to show that, for certain spaces X, the Smyth and Hoare hyperspace constructions commute, namely that QHX and HQX are homeomorphic. The most complete such result is due to Matthew de Brecht and Tatsuji Kawai in 2019; they showed that this is the case exactly when X is consonant. I will give a simplified exposition of their proof, and I will show that essentially the same proof shows that QsHX and HQsX are homeomorphic, for every topological space whatsoever. Read the full post.

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Algebras of filter-related monads: II. KZ-monads

Alan Day and Oswald Wyler once proved that the algebras of the filter monad on the category Top0 of T0 topological spaces are exactly the continuous (complete) lattices. Martín Escardó later gave a very interesting proof of this fact, using a category-theoretic construction due to Anders Kock which he calls KZ-monads. My purpose is to talk about Escardó’s argument; but mostly, really, to put forward his notion of KZ-monad, which is a true categorical gem. Read the full post.

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Algebras of filter-related monads: I. Ultrafilters and Manes’ theorem

In 1969, Ernest Manes proved the following remarkable result: the algebras of the ultrafilter monad on Set are exactly the compact Hausdorff spaces. This is remarkable, because it gives a purely algebraic/category-theoretic of the otherwise purely topological notion of compact Hausdorff spaces. I will explain how this is proved, and I will give a few pointers to some extension results of the same kind. Read the full post.

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A report from ISDT’22: one-step closure; c-spaces are not CCC

I have been attending the 9th International Symposium on Domain Theory (ISDT’22), which took place online, July 4-6, 2022, in Singapore. This was a fine conference indeed, and it ran very smoothly. I initially intended to give a summary of what happened there, but in the end I decided to concentrate on just two contributed papers: one by Hualin Miao, Qingguo Li, and Dongsheng Zhao, about those posets that have one-step closure, namely in which one can compute the Scott-closure of a downwards-closed subset A by just taking the collection of suprema of directed families included in A; the other one by Zhenchao Lyu, Xiaolin Xie, and Hui Kou, who showed that the category of c-spaces and the category of locally finitary compact spaces are not Cartesian-closed, by an argument that is both easy and clever. Read the full post.

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Q is not consonant: the Costantini-Watson argument

I have already given an argument for the non-consonance of the Sorgenfrey line R here. I would now like to explain why the space Q of rational numbers is not consonant either. That is quite a challenge. The most easily accessible proof is due to Costantini and Watson, but it still requires some effort to understand. Fortunately, the topological game of last time will help us make sense of at least one half of the construction. Read the full post.

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Compact scattered subsets and a topological game

Showing that Q is not consonant is quite an ordeal. I have finally managed to understand one of the existing proofs of this fact, due to Costantini and Watson. This would be a bit too long to cover entirely in one post, so the bulk of the explanation will be for another time. Instead, I will explain why the compact subsets of Q are all scattered, and what it means, but the important point of this month’s post is that, reading between the lines, the Costantini-Watson argument relies on a property that I will characterize through the use of a topological game G(K), resembling the strong Choquet game, in which we will see that player I has a winning strategy if K is compact and scattered—and that is an if and only if in any T2 space. Read the full post.

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Topological lattices with small semilattices

I would like to explain a clever counterexample due to Jimmie Lawson in 1970, or rather a slight variant of it, pertaining to the theory of topological semilattices and to a property that crops up naturally, namely having small semilattices. Before I can do this, I will have to spend some time explaining what topological semilattices are, and how small semilattices arise naturally. For motivational purposes, I will consider the problem of characterizing the algebras of the so-called finitary Smyth hyperspace monad, a question that Andrea Schalk has solved, among others, in her 1993 PhD thesis. Read the full post.

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