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Author Archives: jgl
Quotients, colimits of dcpos, and related matters
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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Corecompact+wellfiltered T0=sober locally compact
Last time, I motivated the construction of the wellfilterification 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 corecompact … Continue reading
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Wellfilterifications
Xiaodong Jia once asked the following question: is every corecompact, wellfiltered 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
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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Bchulls and Clathulls
Boundedcomplete domains (bcdomains) are an incredibly useful form of continuous dcpos. Given a continuous dcpo X, is there a completion of X as a bcdomain, for example a free bcdomain on X? That does not exist in general, but Yuri … Continue reading
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Shimrat’s theorem
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
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 firstcountable, 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
Johnstone space J (1981) is the most famous example of a nonsober dcpo. In 1982, Isbell came up with a nonsober 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
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 corecompact, nonlocally compact space
Last time, I had announced that we would do Exercise V5.25 of the red book, constructing a corecompact, yet not locally compact, space. And this is exactly what we shall do: read the full post.
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