| Summary: | 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.
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