Tag Archives: compactness

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

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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

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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

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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

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