On joins of complemented sublocales
A sublocale of a locale \(L\) is a meet-subset \(S\) such that for all \(s \in S\) and for all \( x \in L\), the Heyting exponential \(x\rightarrow s\) belongs to \(S\). In this paper, the author deals with several aspects concerning sublocale lattices, i.e. the natural subobject lattices in the category \(\mathbf{Loc}\) of locales. Sublocales of a given locale \(L\) constitute under the natural inclusion ordering a coframe \(S(L)\), that is, its order-theoretic dual \(S(L)^{op}\) is again a locale. In a recent work [Houston J. Math. 45, No. 1, 21--38 (2019; Zbl 1459.06008)], \textit{J. Picado} et al. show that under a weak point-free separation axiom (namely, subfitness) the subset \(S_c(L)\) of \(S(L)\) consisting of joins of closed sublocales is a Boolean algebra, and moreover it coincides precisely with the Booleanization \(S_b(L)\) of \(S(L)\). Under the above results, the author study \(S_b(L)\) for a general locale \(L\), and show that \(S_c(L)\) is always a subframe of \(S_b(L)\). Moreover, for a \(T_D\)-space \(X\), he prove that \(S_b(\Omega(X))\) is precisely the set of classical subspaces of \(X\), and that a locale \(L\) is \(T_D\)-spatial if and only if the Boolean algebra \(S_b(L)\) is atomic. Finally, given that the Booleanization \(B_L\) of a locale \(L\) is not a functorial construction, the author study some functoriality properties of \(S_b(L)\) in order tu obtain an alternative approach.
scientific article; zbMATH DE number 7441052
scientific article