Revisiting the relation between subspaces and sublocales

Abstract

We revisit results concerning the connection between subspaces of a space and sublocales of its locale of open sets. The approach we present is based on the observation that for every locale $L$ its spatial sublocales $\mathsf{sp}[\mathsf{S}(L)]$ form a coframe which is isomorphic to the coframe $\mathsf{sob}[\mathcal{P}(\mathsf{pt}(L))]$ of sober subspaces of $\mathsf{pt}(L)$. We characterize the frames $L$ such that the spatial sublocales of $\mathsf{S}(L)$ perfectly represent the subspaces of $\mathsf{pt}(L)$. We prove choice-free, weak versions of the results by Niefield and Rosenthal characterizing those frames such that all their sublocales are spatial (i.e., those such that the sober subspaces of $\mathsf{pt}(L)$ perfectly represent the sublocales of $L$). We do so by using a notion of essential prime which does not rely on the existence of enough minimal primes above every element. We will re-prove Simmons' result that spaces such that the sublocales of $Ω(X)$ perfectly represent their subspaces are exactly the scattered spaces. We will characterize scattered spaces in terms of a strong form of essentiality for primes. We apply these characterizations to show that, when $L$ is a spatial frame and a coframe, $\mathsf{pt}(L)$ is scattered if and only if it is $T_D$, and this holds if and only if all the primes of $L$ are completely prime.