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Category Archives: Uncategorized
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 non-consonance 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
Posted in Uncategorized
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 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 … 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
Posted in Uncategorized
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 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 … Continue reading
Posted in Uncategorized
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
Posted in Uncategorized
Tagged compactness, hyperspace, powerdomain
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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 … Continue reading
Posted in Uncategorized
Tagged algebra, continuous lattice, filter, monad
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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 … Continue reading
Posted in Uncategorized
Tagged algebra, compactness, filter, monad
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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 … Continue reading
Posted in Uncategorized
Tagged cartesian closeness, closure, exponentiability
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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 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
Posted in Uncategorized
Tagged compact, consonance, counterexample, game, scattered
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