Joins of closed sublocales

• Main Author: Picado, Jorge
• Publication Date: 2019
• Other Authors: Pultr, Aleš, Tozzi, Anna
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: https://hdl.handle.net/10316/90477
• topic: Frame, locale, sublocale, nucleus, sublocale lattice, coframe, open sublocale, closed sublocale, T1-space, induced subspace, subfit frame, fit frame, Booleanization.

Sublocales that are joins of closed ones constitute a frame S_Vc(L) embedded as a sup-sublattice into the coframe S(L) of sublocales of L. We prove that in the case of subfit L it is a subcolocale of S(L), that it is then a Boolean algebra and in fact precisely the Booleanization of S(L). In case of a T_1-space X, S_Vc(\Omega(X)) picks precisely the sublocales corresponding to induced subspaces. In linear L and more generally if L is also a coframe, S_Vc(L) is both a frame and a coframe, but with trivial exceptions not Boolean and not a subcolocale of S(L).