Author Archives: jgl

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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Borel sets, analytic sets, and the Baire property

Posted on January 20, 2020 by jgl

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 … Continue reading →

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The Rasiowa-Sikorski lemma and the Baire property

Posted on December 23, 2019 by jgl

Merry Christmas! And a Happy New Year, too. There are incredible links between logic and topology, and I would like to start with something called the Rasiowa-Sikorski lemma in logic. That is a theorem that states the existence of certain … Continue reading →

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Quotients, colimits of dcpos, and related matters

Posted on November 20, 2019 by jgl

How do we build colimits in the category Top of topological spaces? This is easy: we take the quotient of a big disjoint sum. How do we build colimits in the category Dcpo of dcpos? This is a much more … Continue reading →

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Core-compact+well-filtered T0=sober locally compact

Posted on October 20, 2019 by jgl

Last time, I motivated the construction of the well-filterification Wf(X) of a space X of X. Xu, Ch. Shen, X. Xi and D. Zhao by saying that it was needed to understand their proof of the fact that every core-compact … Continue reading →

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Well-filterifications

Posted on September 24, 2019 by jgl

Xiaodong Jia once asked the following question: is every core-compact, well-filtered space automatically locally compact? The question was solved positively this year by J. Lawson and X. Xi. I originally planned to try and explain their result. Even more recently, … Continue reading →

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