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Author Archives: jgl
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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Hoover’s maximal limit spaces II: products, liftings, retracts, function spaces, and hyperspaces
Last time, we had introduced Hoover’s maximal limit spaces: spaces in which every convergent filter has a unique largest limit. That notion is closed under many constructions, as we will see: products, liftings, retracts, notably, and that is elementary. The … Continue reading
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Tagged coherence, convergence, filters, function space, hyperspace, sobriety, strong sobriety, weak Hausdorffness
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