Quasi-Uniform Spaces I: Pervin Quasi-Uniformities, Pervin Spaces

A uniform space is a natural generalization of the notion of a metric space, on which completeness still makes sense. It is rather puzzling that I managed to avoid the subject of quasi-uniform spaces in something like the 7 years that this blog existed… and it is time that I started. I will only say very classical things, and I will concentrate one a construction due to William Pervin, simplifying an earlier result of Császár, and which shows that every topological space is quasi-uniformizable. Read the full post.

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On the word topology, and beyond

Today I (Jean G.-L.) have the pleasure to have a guest, Aliaume Lopez. We are going to talk about the word topology on X*. In the book, there is a so-called Topological Higman Lemma that says that, if X is a Noetherian space, then X* is Noetherian in the word topology, generalizing a famous theorem due to Graham Higman. However, there used to be no characterization of the word topology as a universal construction, say as a finest topology or a coarsest topology with some properties. Aliaume has managed to find a satisfactory, and simple, answer to this question. We will then discuss the case of infinite words, and we will end with a conjecture which, if true, would provide us with a large set of new Noetherian spaces. Read the full post.

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Chains and nested spaces

A chain is a totally ordered poset, and a nested space is a topological space whose lattice of open sets is a chain. That may seem like a curious notion, although you might say that the Scott topology on the real line makes it a nested space—so you know that there at least one natural example of the concept. I will show that nested spaces and chains have very strong topological properties. To start with, I will show you why every chain is a continuous poset. I will then tell you how nested spaces arise from the study of so-called minimal Tand TD topologies, as first explored by R. E. Larson in 1969. And I will conclude with a simple proof of a recent theorem by Mike Mislove. Read the full post.

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TD spaces

In any topological space, the closure of any one-element set {x} is also its downward closure ↓x with respect to the specialization preordering. A TD space is a topological space in which, for every point x, ↓x – {x} is closed, too. This seemingly weird concept was introduced by Aull and Thron in a 1962 paper, but it has funny and interesting applications, notably in the comparison of the notions of subspaces and of sublocales, and in Thron’s so-called lattice equivalence problem. I will also mention the Skula topology again… Read the full post.

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Zhao, Xi and Chen’s well-filtered, non-sober dcpo

There are several known examples of dcpos that are well-filtered, but not sober, and I have already mentioned one due to Xiaodong Jia. I would like to explain another one, due to Dongsheng Zhao, Xiaoyong Xi, and Yixiang Chen. This is a very simple modification of Johnstone’s non-sober dcpo J. Contrarily to Xiaodong Jia’s dcpo (and to J), it is uncountable, but it may be easier to see why it must be well-filtered: everything mostly boils down to a cardinality argument, or rather, as I will argue, to the properties of so-called regular ordinals. Read the full post.

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Quasi-Polish spaces as rounded ideal completions

This month, a pearl by Matthew de Brecht. It is known that the rounded ideal completion of an abstract basis (a set B with a transitive, interpolative relation) is a continuous dcpo, and that all continuous dcpos can be obtained this way. What do you get if you remove the requirement of interpolation? Well, and assuming B countable… exactly the quasi-Polish spaces! Read the full post.

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Convergence without points

Can you define convergence without mentioning points? More precisely, is there any form of Stone duality for convergence spaces, instead of just topological spaces? The short answer is yes. For the long answer, read the full post.

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X. Jia’s well-filtered, non-sober dcpo

[Business as usual, despite all viruses!] Peter Johnstone once showed the existence of a dcpo J that is not sober in its Scott topology. That dcpo is not well-filtered either. Is there a dcpo that is not sober but is well-filtered? That is true, and the first one who found an example is Hui Kou. Since then, Xi and Zhao have also given another example, and I would like to describe another example of such a dcpo, due to Xiaodong Jia in his PhD thesis. Both Xi and Zhao’s example and Jia’s example are pretty simple spaces, but X. Jia’s example is countable. Read the full post.

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Dcpos built as graphs of functions

Let X and P be two dcpos, and let ψ be a map from X to P. When is the graph of ψ a dcpo? I will give you a funny sufficient condition, which involves the so-called d-topology, and Hausdorffness. I will briefly explain how this can be used to show that every Π02 subset of a continuous dcpo is domain-complete, namely, is homeomorphic to a Gδ subset of some other continuous dcpo. Read the full post.

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Borel sets, analytic sets, and the Baire property

Let me venture into the realm of σ-algebras. Yes, you might say, that is measure theory, not topology… but topology plays an important role in measure theory and, for that matter, descriptive set theory. I will tell you about sets with the Baire property. Those are pretty simple objects, or at least they appear to be simpler than Borel sets, but we will see that this is the other way around: all Borel sets have the property of Baire. The proof is pretty easy, as well. I will also spend some more time to explain a more complicated result, due to O.M. Nikodým, and which says that all A-sets have the property of Baire as well, in a second-countable space. None of that ever uses any Hausdorffness, or in fact any separation property whatsoever. Read the full post.

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