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 one post, so the bulk of the explanation will be for another time. Instead, I will explain why the compact subsets of Q are all scattered, and what it means, but the important point of this month’s post is that, reading between the lines, the Costantini-Watson argument relies on a property that I will characterize through the use of a topological game G(K), resembling the strong Choquet game, in which we will see that player I has a winning strategy if K is compact and scattered—and that is an if and only if in any T2 space. Read the full post.

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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. Before I can do this, I will have to spend some time explaining what topological semilattices are, and how small semilattices arise naturally. For motivational purposes, I will consider the problem of characterizing the algebras of the so-called finitary Smyth hyperspace monad, a question that Andrea Schalk has solved, among others, in her 1993 PhD thesis. Read the full post.

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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 one may wonder when they coincide. Outside of the realm of posets in their Scott topology, we will see that they rather rarely coincide. The result that is the core of this month’s post is that they do if X is a poset (with its Scott topology) satisfying what I will call the Chen-Kou-Lyu property; and that this property holds if the poset X is core-compact, or first-countable. Read the full post.

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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 paper with Achim Jung. This is a very nice theory, which looks a lot like ordinary Stone duality between topological spaces and frames, but with a few twists. As we will see, the resulting monad, which I would like to call algebraicization, turns every L-domain into an algebraic L-domain. Read the full post.

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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 two applications of the notion. One, which I will actually present last, is a baby version of the so-called Groemer theorem. This baby Groemer theorem is non-trivial, but has an amazingly simple proof, due to Klaus Keimel; we have used it to prove non-Hausdorff generalizations of a line of theorems due to Choquet, Kendall and Matheron. The other application is due to Zhenchao Lyu and Xiaodong Jia. The Smyth powerdomain Q(X) of a space X is locally compact if and only if X is, and they were interested in knowing whether the same would happen with “core-compact” instead of “locally compact”. The answer is no, and this rests on a clever use of irredundancy, and the existence of a core-compact, non-locally compact space. Read the full post.

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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 its associated monad, sheafification. This is all well-known, but then I would like to apply all that to the presheaf of locally monotone functions of a prestream, which we had already started to examine last time. We will obtain a funny structure that I will call stratified étale maps. Read the full post.

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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 to explore what sheaf technology can bring us in the study of streams and prestreams. To start with, I will introduce the classical notion of the étale space of a presheaf, and illustrate that on sheaves of locally monotone (resp., and continuous) maps on streams, and particularly in the case of the directed circle. Read the full post.

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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 for a blog. Instead, I will start a sequence of posts on the notion of stature of a Noetherian space. Since the notion is defined by analogy with a similar notion in the special case of well-partial-orderings, and coincides with something called maximal order types, I will spend this month’s post explaining the latter notion. Read the full post.

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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 functor from Top to Set. In other words, that forgetful functor is topological, but there are many others, including some involving streams, prestreams, and d-spaces. We will see some of the classical properties of topological functors, notably that topological functors are self-dual, and that they preserve and create both limits and colimits. Read the full post.

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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 SD, discovered by Emmanuel Haucourt. S. Krishnan’s purpose was really to talk about streams, not prestreams, and I will show how they emerge from the study of that adjunction. In other words, we will answer the following question partly: what makes the prestreams of the form S(X, dX) so special? Read the full post.

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