2020/10/11 by Anna Laura Suarez, Suarez, Anna Laura
Mathematics · #06D22 #Advanced Topology and Set Theory #Category Theory (math.CT) #FOS: Mathematics #Functional Analysis (math.FA) #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2010.05284
openalex publication_date 2020/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 sp[S(L)] form a coframe which is isomorphic to the coframe sob[P(pt(L))] of sober subspaces of pt(L). We characterize the frames L such that the spatial sublocales of S(L) perfectly represent the subspaces of 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 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, pt(L) is scattered if and only if it is TD, and this holds if and only if all the primes of L are completely prime.