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Category Archives: Uncategorized
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
Posted in Uncategorized
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
Posted in Uncategorized
Tagged cartesian closeness, closure, exponentiability
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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
Posted in Uncategorized
Tagged compact, consonance, counterexample, game, scattered
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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
Posted in Uncategorized
Tagged compact, game, ordinal, scattered
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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
Posted in Uncategorized
Tagged algebra, hyperspace, lattice, monad, powerdomain
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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
Posted in Uncategorized
Tagged corecompactness, firstcountability, hyperspace, powerdomain, Scott topology, Vietoris topology
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Ldomains, 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 socalled Ldomains to socalled distributive Dsemilattices. This was refined later in a common … Continue reading
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Tagged Ldomain, Stone duality
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Irredundant families, the Smyth powerdomain, the LyuJia theorem, and the baby Groemer theorem
A ∩semilattice of sets is a family of sets that is closed under finite intersections, and it is irredundant if and only if all its nonempty elements are irreducible. That sounds like a ridiculously overconstrained notion, but I will give … Continue reading
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Tagged compactness, corecompactness, hyperspace, powerdomain
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Sheaves and streams II: sheafification, and stratified étale maps
In part I, I explained how one can build the étale space of a presheaf F over a topological space X. I will show how one can retrieve a sheaf from an étale map, leading to a nice adjunction and … Continue reading
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Tagged étale map, sheaf, stream
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Sheaves and streams I: sheaves of locally monotone maps
Sheaves are a fundamental notion. In this post and later posts, I would like to explain some of the basic theory of the most mundane notion of sheaves: sheaves of sets over a topological space. My real goal is really … Continue reading
Posted in Uncategorized
Tagged étale map, sheaf, stream
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