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Author Archives: jgl
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, semilattice
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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 core-compactness, first-countability, hyperspace, powerdomain, Scott topology, Vietoris topology
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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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Tagged L-domain, Stone duality
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Irredundant families, the Smyth powerdomain, the Lyu-Jia 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 non-empty elements are irreducible. That sounds like a ridiculously overconstrained notion, but I will give … Continue reading
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Tagged compactness, core-compactness, 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
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Tagged étale map, sheaf, stream
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Statures of Noetherian spaces I: maximal order types of wpos
I have had a very gifted masters 2 student from mid-May to late July, Bastien Laboureix. He mostly solved the questions I had left open here. I wanted to report on his work, but that is a lot too technical … Continue reading
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Tagged noetherian, ordinal, wqo
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Topological functors I: definition, duality, limits and colimits
I have briefly mentioned topological functors in a recent post. It is time for me to explain what they are. This is a truly wonderful concept, which abstracts topological spaces away and concentrates on the key properties of the forgetful … Continue reading
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Tagged category theory, topological functor
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From prestreams to streams
Last time, I had described two satisfactory models of topological spaces with a local direction of time: Marco Grandis’ d-spaces, and Sanjeevi Krishnan’s prestreams. The two kinds form categories that are related by an adjunction S ⊣ D, discovered by … Continue reading