Author Archives: jgl

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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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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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 →

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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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On countability: the compact completed sequence

Posted on May 20, 2019 by jgl

Recently, Matthew de Brecht sent me a proof of a neat and rather surprising result: the product and the Scott topologies coincide for products of first-countable, not necessarily continuous, posets. This rests on a clever argument, inspired by techniques invented … Continue reading →

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Isbell’s non sober complete lattice

Posted on April 21, 2019 by jgl

Johnstone space J (1981) is the most famous example of a non-sober dcpo. In 1982, Isbell came up with a non-sober complete lattice. His construction is so complex that most authors use it as a black box. I would like … Continue reading →

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Countably presented locales

Posted on March 24, 2019 by jgl

Reinhold Heckmann showed the following in a very nice paper of 2014: every countably presented locale is spatial. What makes it even nicer is that he shows how tightly this is connected with the Baire property. This also gives a … Continue reading →

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A core-compact, non-locally compact space

Posted on February 20, 2019 by jgl

Last time, I had announced that we would do Exercise V-5.25 of the red book, constructing a core-compact, yet not locally compact, space.  And this is exactly what we shall do: read the full post.

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