Abstract
A $\sigma$-frame is a poset with countable joins and finite meets in which binary meets distribute over countable joins. The aim of this paper is to show that $\sigma$-frames, actually $\sigma$-locales, can be seen as a branch of Formal Topology, that is, intuitionistic and predicative point-free topology. Every $\sigma$-frame $L$ is the lattice of Lindel\"of elements (those for which each of their covers admits a countable subcover) of a formal topology of a specific kind which, in its turn, is a presentation of the free frame over $L$. We then give a constructive characterization of the smallest (strongly) dense $\sigma$-sublocale of a given $\sigma$-locale, thus providing a "$\sigma$-version" of a Boolean locale. Our development depends on the axiom of countable choice.
Highlights
It is well known that the set B(H) = {x ∈ H | x = − − x} of stable elements of a complete Heyting algebra H is a complete Boolean algebra
The aim of this paper is to show that σ-frames, σ-locales, can be seen as a branch of Formal Topology, that is, intuitionistic and predicative point-free topology
Every σ-frame L is the lattice of Lindelof elements of a formal topology of a specific kind which, in its turn, is a presentation of the free frame over L
Summary
From the point of view of the category of locales, this means that every locale L contains a Boolean sublocale B(L), which can be characterized as the smallest dense sublocale of L. One of the main advantages of his approach is that powersets are examples of overlap algebras ( they are precisely the atomic ones), they are not Boolean, constructively. It has recently turned out [Cir16] (see [CC20]) that overlap algebras can be understood as the smallest strongly dense sublocales (in the sense of [Joh89]) of overt locales. Key words and phrases: Formal Topology, σ-frames, overlap algebras, overt locales, strongly dense sublocales
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