Category Archives: Uncategorized

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

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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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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

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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

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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

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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

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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 … Continue reading

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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. … Continue reading

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When do the upper (a.k.a., lower Vietoris) and Scott topologies coincide on the Hoare hyperspace of a space?

I would like to talk about a nifty, recent result due to Yu Chen, Hui Kou, and Zhenchao Lyu. There are two natural topologies on the Hoare hyperspace of a space X, the Scott and the lower Vietoris topology, and … Continue reading

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L-domains, stable open sets, and stable Stone duality

Stone duality relates topological spaces and locales (or frames). But there are really many sorts of Stone dualities. In 1997, Yixiang Chen studied Stone dualities that relate so-called L-domains to so-called distributive D-semilattices. This was refined later in a common … Continue reading

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