Joins of Closed Sublocales are not Always A Coframe

Given a locale L , the ordered collection Sc(L) S c ( L ) of joins of closed sublocales forms a frame—somewhat unexpectedly, as it is naturally embedded in the coframe of all sublocales of L , where by coframe we mean the order-theoretic dual of a frame. This construction has attracted attention in point-free topology: as a maximal essential extension in the category of frames, for its (non-)functorial properties, its relation to canonical extensions and exact filters of frames, etc. A central open question of the theory, posed by Picado, Pultr, and Tozzi in 2019, asked whether Sc(L) S c ( L ) is always a coframe, or whether there exists a locale for which this fails. In this paper, we resolve this question in the negative by constructing a locale L such that Sc(L) S c ( L ) is not a coframe. The main challenge in such question lies in the difficulty of understanding exact infima in Sc(L) S c ( L ) ; we circumvent this by analysing a certain separation property satisfied by Sc(L) S c ( L ) .