Tag Archives: sober space

Bitopological spaces, d-frames, and Jung-Moshier duality

Posted on June 20, 2025 by jgl

Stone duality is an adjunction between the category Top of topological spaces and the category Loc of locales, namely the opposite of the category Frm of frames. Is there a similar-looking adjunction between the category biTop of bitopological spaces—namely, sets … 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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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 →

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Shimrat’s theorem

Posted on June 24, 2019 by jgl

Marcus Tressl just found a horrible mistake in the book: I had claimed that the T0 quotient of the topological quotient of any sober space by any equivalence relation is sober, but that is completely wrong. In fact, Moshe Shimrat … Continue reading →

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