Category Archives: Uncategorized

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.

Posted in Uncategorized | Tagged convergence, duality, filters, Stone duality | Comments Off

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 →

Posted in Uncategorized | Tagged counterexample, sober space, well-filtered space | Comments Off

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 →

Posted in Uncategorized | Tagged d-topology, dcpo, domain-complete, quasi-polish space | Comments Off

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 →

Posted in Uncategorized | Tagged analytic set, Baire property, Borel set | Comments Off

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 →

Posted in Uncategorized | Tagged Baire property, filters, logic | Comments Off

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 →

Posted in Uncategorized | Tagged dcpo, quotient | Comments Off

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 →

Posted in Uncategorized | Tagged core-compactness, sober space, well-filtered space | Comments Off

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 →

Posted in Uncategorized | Tagged well-filtered space | Comments Off

Sober subspaces and the Skula topology

Posted on August 22, 2019 by jgl

It often happens that one wishes to show that a certain subspace A of a given sober space X is sober. The following is a pearl due to Keimel and Lawson, which was mentioned to me by Zhenchao Lyu in … Continue reading →

Posted in Uncategorized | Tagged skula topology, sober space | Comments Off

Bc-hulls and Clat-hulls

Posted on July 20, 2019 by jgl

Bounded-complete domains (bc-domains) are an incredibly useful form of continuous dcpos. Given a continuous dcpo X, is there a completion of X as a bc-domain, for example a free bc-domain on X? That does not exist in general, but Yuri … Continue reading →

Posted in Uncategorized | Tagged bounded-complete dcpo, complete lattice | Comments Off