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Tag Archives: compactness
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
Posted in Uncategorized
Tagged coherence, compactness, counterexample, patch topology, well-filtered space
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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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Compact semilattices without small semilattices I: interval homomorphisms, products, and the Hoare hyperspace
I have already talked about compact semilattices before, but there is a lot more to say, especially on the subject of having small semilattices or not. Zhenchao Lyu is joining me this month, and we will pursue this next month. … Continue reading
Posted in Uncategorized
Tagged compactness, continuous dcpo, Hoare hyperspace, powerdomain, semilattice, Urysohn
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Strongly compact sets and the double hyperspace construction
The notion of strongly compact set is due to Reinhold Heckmann. A few months ago, I said that I would explain why the sobrification of the space Qfin(X) of finitary compact sets on a sober space X is not the … Continue reading
Posted in Uncategorized
Tagged compactness, hyperspace, powerdomain
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Algebras of filter-related monads: I. Ultrafilters and Manes’ theorem
In 1969, Ernest Manes proved the following remarkable result: the algebras of the ultrafilter monad on Set are exactly the compact Hausdorff spaces. This is remarkable, because it gives a purely algebraic/category-theoretic of the otherwise purely topological notion of compact … Continue reading
Posted in Uncategorized
Tagged algebra, compactness, filter, monad
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Irredundant families, the Smyth powerdomain, the Lyu-Jia theorem, and the baby Groemer theorem
A ∩-semilattice of sets is a family of sets that is closed under finite intersections, and it is irredundant if and only if all its non-empty elements are irreducible. That sounds like a ridiculously overconstrained notion, but I will give … Continue reading
Posted in Uncategorized
Tagged compactness, core-compactness, hyperspace, powerdomain
Comments Off on Irredundant families, the Smyth powerdomain, the Lyu-Jia theorem, and the baby Groemer theorem