Irredundant families, the Smyth powerdomain, the Lyu-Jia theorem, and the baby Groemer theorem

Posted on January 20, 2022 by jgl

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

Posted on December 20, 2021 by jgl

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

Posted on November 22, 2021 by jgl

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

Posted on October 23, 2021 by jgl

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

Posted on September 22, 2021 by jgl

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

Posted on August 24, 2021 by jgl

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 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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Prestreams and d-spaces

Posted on July 25, 2021 by jgl

How do you model a topological space with a direction of time? That should seem easy; for example, a topological space with a preordering should be enough. But how do you model the directed circle, where times goes counterclockwise? That should seem harder. Fortunately, there are at least two satisfactory models: Marco Grandis’ d-spaces, and Sanjeevi Krishnan’s (pre)streams, which I will describe. Both appeared in 2009. Although they look rather different, they are connected by an idempotent adjunction, which restricts to an equivalence between so-called complete d-spaces and Haucourt streams, as discovered by Emmanuel Haucourt in 2012. Read the full post.

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First-countable spaces and their Smyth powerdomain

Posted on June 22, 2021 by jgl

This month, we will look at certain conditions recently found by He, Li, Xi and Zhao in 2019, and then by Xu and Yang in 2021, in order to ensure that the Smyth powerdomain Q(X) (with the Scott topology) of a space X is a dcpo model of X, and in order to ensure that the Scott and upper Vietoris topologies agree on Q(X)… without the usual condition of local compactness, but under some first-countability conditions. I got interested in this while Xiaodong Jia and I explored the Smyth powerdomain of the Sorgenfrey line. Read the full post.

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The Sorgenfrey line is not consonant

Posted on May 20, 2021 by jgl

In Exercise 5.4.12 of the book, I ask the reader to prove that neither the space of rationals, Q, nor the Sorgenfrey line, Rℓ, is consonant. But the proofs I had in mind were much too simple-minded to stand any chance of succeeding, hence I classified this as important blooper #5 in the list of errata. Good news: Showing that Rℓ is not consonant is not that hard, finally. I will explain the argument in the full post. This will also be an excuse to explain some additional topological properties of Rℓ, an introduction to hereditary Lindelöfness (we will see that Rℓ is hereditarily Lindelöf, although it is not second-countable), and a few additional things in the appendices.

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Bitopological spaces and stable compactness

Posted on April 22, 2021 by jgl

A while back (in March 2019, to be precise), Tomáš Jakl told me that he had a nice, short proof of the fact that the categories of stably compact spaces (and perfect maps) and compact pospaces (and continuous order-preserving maps) are equivalent. He uses an approach through bitopological spaces, and this will give me an opportunity to talk about them. Just as with quasi-uniform spaces, I cannot believe it took me so much time before I mentioned bitopological spaces! Read the full post.

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