Category Archives: Uncategorized

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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Convergence without points

Posted on April 20, 2020 by jgl

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

Posted on March 20, 2020 by jgl

[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 … Continue reading →

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Dcpos built as graphs of functions

Posted on February 20, 2020 by jgl

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. … Continue reading →

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