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Category Archives: Uncategorized
Posets determined by countably many corecompact 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
Posted in Uncategorized
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
Posted in Uncategorized
Tagged colimit, corecompactness, 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
Posted in Uncategorized
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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
Posted in Uncategorized
Tagged conformity, consonance, maximal limit space, Scott topology, Smyth hyperspace, sobriety, weak Hausdorffness
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On products of dcpos, the MiaoXiLiZhao 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
Posted in Uncategorized
Tagged corecompactness, countability, firstcountability, 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
Posted in Uncategorized
Tagged coherence, convergence, filters, function space, hyperspace, sobriety, strong sobriety, weak Hausdorffness
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Hoover’s maximal limit spaces I: local strong sobriety, bounded supcompleteness and weak Hausdorffness
In 1995, Douglas Hoover introduced and studied a notion of maximal limit spaces: spaces in which every convergent net has a unique largest limit. This has connections with many other kinds of spaces that we have explored already, and I … Continue reading
Posted in Uncategorized
Tagged coherence, convergence, filters, sobriety, strong sobriety, weak Hausdorffness
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Skula spaces II: the DowWatson counterexample
Last time, we have started to explain some results due to A. Dow and S. Watson, and we have seen that every compact T0 scattered space of scattering height at most 3 is Skula, namely can be obtained from a … Continue reading
Posted in Uncategorized
Tagged scattered, skula topology
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Skula spaces I: clopen selectors
A Skula space is a space that is obtained from another space X by giving it the Skula topology instead, which is generated by the open sets and the closed sets of the original space X. In 1990, Alan Dow … Continue reading
Posted in Uncategorized
Tagged compactness, connectedness, noetherian, scattered, skula topology
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The (positive) unique structure theorem
There is an amazing theorem in the theory of analytic spaces, called the unique structure theorem: any two countable families F1 and F2 of Borel subsets of an analytic space X that separate the points of X generate the same … Continue reading
Posted in Uncategorized
Tagged Borel set, omegacontinuous dcpo, quasianalytic space, quasipolish space, sigmaalgebra, sigmalattice
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