Category Archives: Uncategorized

Quasi-continuous domains and the Smyth powerdomain

Posted on February 22, 2021 by jgl

“Quasi-continuous domains and the Smyth powerdomain” is the title of a very nice 2013 paper by Reinhold Heckmann and Klaus Keimel. I will not talk about quasi-continuous domains in this post. Rather, I will mention three pearls that this paper … Continue reading →

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Quasi-uniform spaces IV: Formal balls—a proposal

Posted on January 23, 2021 by jgl

Formal balls are an extraordinarily useful notion in the study of quasi-metric, and even hemi-metric spaces. Is there any way of extending the notion to the case of quasi-uniform spaces? This is what I would like to start investigating. This … Continue reading →

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Quasi-uniform spaces III: Smyth-completeness, symcompactness

Posted on December 20, 2020 by jgl

We embark on the study of notions of completeness for quasi-uniform spaces, and we concentrate on Smyth-completeness. We will see that at least two familiar theorems from the realm of quasi-metric spaces generalize to quasi-uniform spaces: all Smyth-complete quasi-uniform spaces … Continue reading →

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Quasi-uniform spaces II: Stably compact spaces

Posted on November 21, 2020 by jgl

There is a standard result in the theory of uniform spaces that shows (again) how magical compact Hausdorff space can be: for every compact Hausdorff space X, there is a unique uniformity that induces the topology of X, and its … Continue reading →

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

Posted on October 19, 2020 by jgl

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 … Continue reading →

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On the word topology, and beyond

Posted on September 20, 2020 by jgl

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 … Continue reading →

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Chains and nested spaces

Posted on August 23, 2020 by jgl

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 … Continue reading →

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TD spaces

Posted on July 20, 2020 by jgl

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 … Continue reading →

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

Posted on June 20, 2020 by jgl

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 … Continue reading →

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Quasi-Polish spaces as rounded ideal completions

Posted on May 20, 2020 by jgl

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 … Continue reading →

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