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Author Archives: jgl
Localic products and Till Plewe’s game
Products in the category of locales resemble, but do not coincide with products in the category of topological spaces. Till Plewe has a nice explanation to this, as I will explain in this month’s post: the localic product of two … Continue reading
Posted in Uncategorized
Tagged consonance, counterexample, frame, game, locale, ordinal
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On Till Plewe’s game and Matthew de Brecht’s nonconsonance arguments
Last time I mentioned that S0 is not consonant. I will give Matthew de Brecht’s proof of that. Perhaps the most interesting part of this proof is a criterion that he proves and uses: if a space X is consonant, … Continue reading
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Tagged consonance, counterexample, game, ordinal
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The space S0
S0 is a space that occurs in Matthew de Brecht’s generalized Hurewicz theorem for quasiPolish spaces, published in 2018. S0 is very simple: it is an infinite countablybranching tree, and if you order it so that the root is at … Continue reading
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 … Continue reading
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Tagged noetherian, ordinal, wqo
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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 socalled locally strongly sober spaces (which, I am ashamed to say, I had heard about but completely forgotten about). At the same time, I was … Continue reading
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Tagged filters
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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 … Continue reading
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Tagged compactness, hyperspace, powerdomain
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Algebras of filterrelated monads: II. KZmonads
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 categorytheoretic construction … Continue reading
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Tagged algebra, continuous lattice, filter, monad
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Algebras of filterrelated 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/categorytheoretic of the otherwise purely topological notion of compact … Continue reading
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Tagged algebra, compactness, filter, monad
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A report from ISDT’22: onestep closure; cspaces are not CCC
I have been attending the 9th International Symposium on Domain Theory (ISDT’22), which took place online, July 46, 2022, in Singapore. This was a fine conference indeed, and it ran very smoothly. I initially intended to give a summary of … Continue reading
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Tagged cartesian closeness, closure
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Q is not consonant: the CostantiniWatson argument
I have already given an argument for the nonconsonance of the Sorgenfrey line Rℓ 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 … Continue reading
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Tagged compact, consonance, counterexample, game, scattered
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