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Author Archives: jgl
Arcs and arc chains II: every arc is an arc chain
Last time, we have seen that every arc chain is an arc. This month, we will see that every arc is an arc chain. The proof proceeds by inventing a suitable partial ordering, the cut-point ordering. We will also see … Continue reading
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Arcs and arc chains
Jimmie Lawson had presented some results about chains in partially ordered spaces at the ISDT conference in 2022, which I wanted to talk about. But this requires Koch’s arc theorem, so I decided to talk about the latter first. That … Continue reading
When is ℒ(X) locally compact? II. The non-consonant case and the non-second-countable case
Last month, we have seen that ℒ(X) is locally compact in its Scott topology (equivalently, core-compact, equivalently, stably compact) if and only if OX is locally compact in its Scott topology (equivalently, core-compact, equivalently, stably compact), and that those properties … Continue reading
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When is ℒ(X) locally compact?
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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Tagged consonance, countability, Hoare hyperspace, Scott topology, Smyth hyperspace
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Compact closed subsets in the patch topology
Given a coherent, well-filtered space X, one can give a pretty explicit description of the compact closed sets in Xpatch, the space obtained by equipping X with its patch topology—and we do not need local compactness or compactness. They are … Continue reading
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Tagged coherence, compactness, counterexample, patch topology, well-filtered space
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Posets determined by countably many core-compact subspaces
Let me return to a topic that I have addressed a few times in recent posts: under which conditions does the Scott topology of the product of two posets coincide with the product topology of the Scott topologies? We have … Continue reading
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Tagged colimit, dcpo, product
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When do products distribute over colimits in Top?
In a recent paper, Lawson and Xu give a new class of posets on which the Scott topology of the (poset) product of two posets coincides with the product topology (of each poset with its Scott topology). I will explain … Continue reading
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Tagged colimit, core-compactness, counterexample, filters, product
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Happy summer holidays 2024!
I will not post anything this month, sorry: I am on holidays, starting in a few days, and for about one month. Said otherwise: I have tried finding some time to write up on something before leaving off, but I … Continue reading
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The complete lattice Lfan (part II)
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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Tagged conformity, consonance, maximal limit space, Scott topology, Smyth hyperspace, sobriety, weak Hausdorffness
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On products of dcpos, the Miao-Xi-Li-Zhao lemma, and the complete lattice Lfan (part I)
The product of a poset P with itself can be given two topologies: the Scott topology of the product, or the product of the Scott topologies. Those two topologies differ in general, but they coincide when P is a continuous … Continue reading
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Tagged core-compactness, countability, first-countability, Hoare hyperspace, ideals, Scott topology
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