Joins of Closed Sublocales are not Always A Coframe

Abstract 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).