σ-locales and Booleanization in Formal Topology

01/29/2018

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by   Francesco Ciraulo, et al.

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A σ-frame is a poset with countable joins and finite meets in which binary meets distribute over countable joins. Aim of this paper is to present σ-frames, actually σ-locales, as a branch of Formal Topology, that is, intuitionistic and predicative pointfree topology. We show that every σ-frame L can be presented as the lattice of Lindelöf elements (those whose covers admit countable subcovers) of a formal topology of a specific kind which, in its turn, corresponds to the free frame over L. We then give a constructive characterization of the smallest dense σ-sublocale of a given σ-locale, thus providing a "σ-version" of a Boolean locale.