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.

Disclosure

“BK20241086, BK20241087). Declaration of interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work, the authors used ChatGPT (OpenAI) to assist with language refinement, manuscript organization, and literature review. After using”

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2026-08-23T05:10:47.981066Z

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