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.
翻译:$\ sigma$- frame 是一个假象, 配有可计算连号, 和可计算连号的有限会议。 本文的目的是显示 $\ sigma$- frames, 实际上是 $\ grama$- clobes, 可以被视为正规地形学的一个分支, 即直观和预设的零点地形学。 每个$\ grama$- frames $- frames, 是 一种特定类型正式的地形学( 其中每个封面都包含可计算子封面的) 的拉特 。 而该表则展示自由框架超过 $。 然后我们给一个特定 $\ sigma$- sublociety 的最小( 粗密度) $\ sigma$- sublocie, 从而给 Boolean 本地提供一个“ $\ sigma$- version ” 。 我们的发展取决于可计算选择的xion 。