On the relation between subspaces and sublocales

Abstract

We present a new approach for investigating the relation between subspaces and sublocales, and use it to find new results as well as new proofs of known results. The approach we present is based on the notion of sublocale as a concrete subcollection of a locale. In particular we show that for a frame L we have that the collection of spatial sublocales sp [ S ( L ) ] is a subcolocale of the coframe S ( L ) . We also re-prove that the collection sob [ P ( pt ( L ) ) ] is a subcoframe of the powerset P ( pt ( L ) ) . We show that the two coframes sp [ S ( L ) ] and sob [ P ( pt ( L ) ) ] are isomorphic. We base our analysis on these facts. 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. 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 give a new proof of Simmons’ result that spaces such that the sublocales of Ω ( X ) perfectly represent their subspaces are exactly the scattered spaces. We characterize scattered spaces algebraically, in terms of a strong form of essentiality for primes, absolute essentiality . 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 T D , and this holds if and only if all the primes of L are completely prime.