Joins of closed sublocales are not always a coframe

Abstract

Given a locale $L$, the collection ${\mathsf{S}}_c(L)$ 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 ${\mathsf{S}}_c(L)$ 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 ${\mathsf{S}}_c(L)$ is not a coframe. The main challenge in such questions lies in the difficulty of understanding exact infima in ${\mathsf{S}}_c(L)$; we circumvent this by analysing a certain separation property satisfied by ${\mathsf{S}}_c(L)$.