## Francesco Ciraulo - $\sigma$-locales in Formal Topology

lmcs:4244 - Logical Methods in Computer Science, January 12, 2022, Volume 18, Issue 1 - https://doi.org/10.46298/lmcs-18(1:7)2022
$\sigma$-locales in Formal Topology

Authors: Francesco Ciraulo

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 Lindelöf 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.

Volume: Volume 18, Issue 1
Published on: January 12, 2022
Accepted on: November 4, 2021
Submitted on: January 30, 2018
Keywords: Mathematics - Logic,Computer Science - Logic in Computer Science,06D22, 03F65
Fundings :
Source : OpenAIRE Graph
• Computing with Infinite Data; Funder: European Commission; Code: 731143

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