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 Q_fin(X) of finitary compact sets on a sober space X is not the Smyth hyperspace Q(X), rather its subspace of strongly compact saturated sets Q_s(X). This what I will start with. I will then present a funny other case where strongly compact sets are required. There is a long line of research purporting to show that, for certain spaces X, the Smyth and Hoare hyperspace constructions commute, namely that QHX and HQX are homeomorphic. The most complete such result is due to Matthew de Brecht and Tatsuji Kawai in 2019; they showed that this is the case exactly when X is consonant. I will give a simplified exposition of their proof, and I will show that essentially the same proof shows that Q_sHX and HQ_sX are homeomorphic, for every topological space whatsoever. Read the full post.