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 two applications of the notion. One, which I will actually present last, is a baby version of the so-called Groemer theorem. This baby Groemer theorem is non-trivial, but has an amazingly simple proof, due to Klaus Keimel; we have used it to prove non-Hausdorff generalizations of a line of theorems due to Choquet, Kendall and Matheron. The other application is due to Zhenchao Lyu and Xiaodong Jia. The Smyth powerdomain Q(X) of a space X is locally compact if and only if X is, and they were interested in knowing whether the same would happen with “core-compact” instead of “locally compact”. The answer is no, and this rests on a clever use of irredundancy, and the existence of a core-compact, non-locally compact space. Read the full post.