形式拓扑中的$\sigma$-locale
$σ$-locales in Formal Topology
AI总结：
本文证明$\sigma$-框架（即$\sigma$-locale）可纳入形式拓扑框架，并构造性地刻画最小强稠密$\sigma$-子locale，给出布尔locale的$\sigma$-版本，依赖可数选择公理。
摘要：
$\sigma$-框架是一个具有可数并和有限交的偏序集，其中二元交对可数并可分配。本文旨在表明$\sigma$-框架，实际上是$\sigma$-locale，可以被视为形式拓扑的一个分支，即直觉主义的、可预测的无点拓扑。每个$\sigma$-框架$L$都是某种特定形式拓扑的Lindelöf元素（即其每个覆盖都有可数子覆盖的元素）之格，而这种形式拓扑反过来又是$L$上的自由框架的一个表示。我们随后给出了给定$\sigma$-locale的最小（强）稠密$\sigma$-子locale的构造性刻画，从而提供了一个布尔locale的“$\sigma$-版本”。我们的发展依赖于可数选择公理。
英文摘要：
A $σ$-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 $σ$-frames, actually $σ$-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 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 $σ$-sublocale of a given $σ$-locale, thus providing a "$σ$-version" of a Boolean locale. Our development depends on the axiom of countable choice.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
