On joins of complemented sublocales

Abstract

The system ${\mathcal {S}}_c(L)$ consisting of joins of closed sublocales of a locale L is known to be a frame, and for L subfit it coincides with the Booleanization ${\mathcal {S}}_b(L)$ of the coframe of sublocales of L. In this paper, we study ${\mathcal {S}}_b(L)$ for a general locale L. We show that ${\mathcal {S}}_c(L)$ is always a subframe of ${\mathcal {S}}_b(L)$. Moreover, if X is a $T_D$-space, we prove that ${\mathcal {S}}_b(\Omega (X))$ is precisely the set of classical subspaces of X, and that a locale L is $T_D$-spatial iff the Boolean algebra ${\mathcal {S}}_b(L)$ is atomic. Some functoriality properties of ${\mathcal {S}}_b(L)$ are also studied.