Joins of closed sublocales are not always a coframe
Igor Arrieta
TL;DR
The paper resolves a long-standing open question by showing that the collection $\mathsf{S}_c(L)$ of joins of closed sublocales of a locale $L$ need not be a coframe. The authors adopt a separation-property approach, constructing a non-spatial frame from $K(L)$ and a regular open set in $\Omega(\mathbb{R})$ to produce a locale $L$ with $\mathsf{S}_c(L)$ failing coframeness, sidestepping the direct computation of exact infima. A concrete counterexample is provided via $\mathsf{S}_c(K(\Omega(\mathbb{R})))$, and the work links this phenomenon to notions of weak subfitness and to symmetric locales. The results illuminate the limits of $\mathsf{S}_c(L)$ as a discretization/extension mechanism in point-free topology and suggest further exploration of how localic separation axioms interact with frame/coframe structures, especially in the space–locale correspondence.
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 $\mathsf{S}_c(L)$.