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Tag Archives: counterexample
Localic products and Till Plewe’s game
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
Posted in Uncategorized
Tagged consonance, counterexample, frame, game, locale, ordinal
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On Till Plewe’s game and Matthew de Brecht’s non-consonance arguments
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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Tagged consonance, counterexample, game, ordinal
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The space S0
S0 is a space that occurs in Matthew de Brecht’s generalized Hurewicz theorem for quasi-Polish spaces, published in 2018. S0 is very simple: it is an infinite countably-branching tree, and if you order it so that the root is at … Continue reading
Q is not consonant: the Costantini-Watson argument
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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Tagged compact, consonance, counterexample, game, scattered
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The Sorgenfrey line is not consonant
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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Tagged consonance, counterexample, powerdomain, valuation
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Plotkin’s powerdomain and the hedgehog
There are three classical powerdomains in domain theory, named after Hoare, Smyth, and Plotkin. The first two are natural and well studied, and the third one is intricate and intriguing. To start with, there are several possible definitions for a … Continue reading
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Tagged counterexample, powerdomain
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Zhao, Xi and Chen’s well-filtered, non-sober dcpo
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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Tagged counterexample, sober space, well-filtered space
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X. Jia’s well-filtered, non-sober dcpo
[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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Tagged counterexample, sober space, well-filtered space
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