Sobriety of Regular Open Algebras in Second-Countable T3 Spaces

Abstract

For a topological space X, let RO(X) be the complete Boolean algebra of regular open subsets of X, ordered by inclusion. We prove that, for every second-countable T3 space X, the Scott space of RO(X) is sober if and only if the set of all isolated points of X is dense in X. Consequently, RO$(\mathbb R^n)$ is not sober for every positive integer $n$. In particular, RO$(\mathbb R)$ is not sober, which provides an answer to an open problem concerning the sobriety of complete Boolean algebras. This characterization also yields a systematic way to obtain more natural examples of complete lattices whose Scott spaces are non-sober.