Tag Archives: consonance

When is ℒ(X) locally compact?

Posted on November 21, 2024 by jgl

Let ℒ(X) be the dcpo of all lower semicontinuous maps from a space X to R+ ∪ {∞}. We give it the Scott topology. Under which conditions on X is it locally compact? We will see that this happens exactly … Continue reading →

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The complete lattice Lfan (part II)

Posted on June 20, 2024 by jgl

Last time, we had started to study the complete lattice Lfan, namely just N × N, with equality as ordering on the first component and the usual ordering on the second component, plus an extra bottom element ⊥, plus an … Continue reading →

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Localic products and Till Plewe’s game

Posted on March 20, 2023 by jgl

Products in the category of locales resemble, but do not coincide with products in the category of topological spaces. Till Plewe has a nice explanation to this, as I will explain in this month’s post: the localic product of two … Continue reading →

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On Till Plewe’s game and Matthew de Brecht’s non-consonance arguments

Posted on February 20, 2023 by jgl

Last time I mentioned that S0 is not consonant. I will give Matthew de Brecht’s proof of that. Perhaps the most interesting part of this proof is a criterion that he proves and uses: if a space X is consonant, … Continue reading →

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Q is not consonant: the Costantini-Watson argument

Posted on June 20, 2022 by jgl

I have already given an argument for the non-consonance of the Sorgenfrey line Rℓ here. I would now like to explain why the space Q of rational numbers is not consonant either. That is quite a challenge. The most easily … Continue reading →

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The Sorgenfrey line is not consonant

Posted on May 20, 2021 by jgl

In Exercise 5.4.12 of the book, I ask the reader to prove that neither the space of rationals, Q, nor the Sorgenfrey line, Rℓ, is consonant. But the proofs I had in mind were much too simple-minded to stand any … Continue reading →

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