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Meta
Tag Archives: sober space
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
Posted in Uncategorized
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
Posted in Uncategorized
Tagged counterexample, sober space, well-filtered space
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Core-compact+well-filtered T0=sober locally compact
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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Tagged core-compactness, sober space, well-filtered space
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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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Tagged skula topology, sober space
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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
Posted in Uncategorized
Tagged Hausdorff space, quotient, sober space
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