Skula spaces II: the Dow-Watson counterexample

Last time, we have started to explain some results due to A. Dow and S. Watson, and we have seen that every compact T0 scattered space of scattering height at most 3 is Skula, namely can be obtained from a (necessarily Noetherian) space X by giving it its Skula topology instead. Today I will explain an example, also due to Dow and Watson, of a compact T0 scattered space of scattering height 4 that is not Skula. Read the full post.

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Skula spaces I: clopen selectors

A Skula space is a space that is obtained from another space X by giving it the Skula topology instead, which is generated by the open sets and the closed sets of the original space X. In 1990, Alan Dow and Stephen Watson gave a simple necessary and sufficient condition for a compact Hausdorff space to be Skula. I will show what it is—it involves the notion of clopen selectors in the title—, how it relates to the theory of Noetherian spaces, and I will end with a corollary of theirs about compact Hausdorff scattered spaces of scattering height at most 3 being Skula. That will be an opportunity to talk about scattered spaces, totally disconnected spaces, connected and quasi-connected components. I would also like start the full post with a tribute to Maurice Pouzet, a nice human fellow and a great mathematician, who passed away on December 31st, 2023.

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The (positive) unique structure theorem

There is an amazing theorem in the theory of analytic spaces, called the unique structure theorem: any two countable families F1 and F2 of Borel subsets of an analytic space X that separate the points of X generate the same σ-algebra. F. Clerc, N. Fijalkow, B. Klin and P. Panangaden used a generalization of this in a 2019 paper about labelled Markov processes, which they call the positive unique structure theorem. It too, applies to any analytic space, and I will show that it even generalizes to a larger class of spaces, which I will call quasi-analytic; those are the continuous images of M. de Brecht’s quasi-Polish spaces (inside an arbitrary topological space, not necessarily Hausdorff or even T0). The treatment I will give is a mix of measure-theoretic and domain-theoretic techniques. Read the full post.

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Compact semilattices without small semilattices II: Gierz’s counterexample

We pursue last month’s post, written with Zhenchao Lyu, and we describe Gierz’s example of a compact semilattice — namely, a compact Hausdorff topological semilattice — which does not have small semilattices. It is a bit simpler than Jimmie Lawson’s counterexample, which was already described on this blog. Read the full post.

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Compact semilattices without small semilattices I: interval homomorphisms, products, and the Hoare hyperspace

I have already talked about compact semilattices before, but there is a lot more to say, especially on the subject of having small semilattices or not. Zhenchao Lyu is joining me this month, and we will pursue this next month. We start with Jimmie Lawson’s characterization of those compact semilattices that have small semilattices as being those that have enough interval homomorphisms. We will then exploit that and show that, once we know just one compact semilattice with small semilattices (for example, one given by J. Lawson himself), then we can produce a whole lot of them, using products or the Hoare hyperspace construction. Read the full post.

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Scott’s formula

There is a well-known formula in domain theory, which, given a monotonic map f from a basis B of a continuous poset X to a dcpo Y, produces the largest continuous map f’ defined on the whole of X and below f on B. This is Scott’s formula: f’(x) ≝ supbB, bx f(b). Generalizing this to the setting where X is a c-space, I will explain when exactly f’ coincides with f on B, hence when f’ is a continuous extension of f. Also, using a motivation from the theory of continuous valuations, I will describe conditions under which any algebraic laws satisfied by f transfer to f’. Read the full post.

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The fundamental theorem of compact semilattices

Bounded-complete domains, or bc-domains, are an amazingly rich kind of continuous domains. They form a Cartesian-closed category, and they are the densely injective topological spaces, among other properties. One characterization of bc-domains which I have not included in the book is that they are related in a very precise sense to Lawson semilattices, namely to compact semilattices with small semilattices. This is the fundamental theorem of compact semilattices, which I will (re)prove by relying a lot on the theory of compact pospaces and stably compact spaces: see the full post.

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Exponentiable locales II: the exponentiable locales are the continuous frames

Two months ago, we have seen that every exponentiable locale had to be continuous, as a frame. We will see that the converse holds: the continuous frames are exactly the exponentiable locales. The result is due to Martin Hyland in 1979. Read the full post.

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The Banaschewski-Lawson-Ershov observation on separate vs. joint continuity

Joint continuity is a stronger property than separate continuity. In what cases are those properties equivalent? The question was solved, partially, by Yuri Ershov in 1997, and completely by Bernhard Banaschewski in 1977 (apparently with a gap in the proof) and by Jimmie Lawson in 1985. The answer has to do with locally finitary compact spaces: read the full post.

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Exponentiable locales I: every exponentiable locale is continuous

The exponentiable objects of Top are exactly the core-compact spaces. Through Stone duality, the core-compact spaces are related to the continuous frames. So here is a wild guess: would the exponentiable locales be exactly the continuous frames? That is indeed true, as was proved by Martin Hyland in 1979 (published in 1981). I will concentrate on one half of the this result for this time, and I will explain why every exponentiable locale must be a continuous dcpo. The proof is very close to the similar result in Top, but, as usual, locales are so much more abstract that similar arguments tend to be harder to understand in Loc; I will do my best. Read the full post.

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