On joins of complemented sublocales

The system 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 S-b(L) of the coframe of sublocales of L. In this paper, we study S-b(L) for a general locale L. We show that S-c(L) is always a subframe of S-b(L). Moreover, if X is a T-D-space, we prove that 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 S-b(L) is atomic. Some functoriality properties of S-b(L) are also studied.