Tag Archives: well-filtered space

Compact closed subsets in the patch topology

Posted on October 20, 2024 by jgl

Given a coherent, well-filtered space X, one can give a pretty explicit description of the compact closed sets in Xpatch, the space obtained by equipping X with its patch topology—and we do not need local compactness or compactness. They are … 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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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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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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