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 notion of sublocale as a concrete subcollection of a locale. 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 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 T D , and this holds if and only if all the primes of L are completely prime.

The coframe of D-sublocales of a locale and the $T_D$ duality

“…We may also ask when it is that the right vertical arrow is an isomorphism. The frames for which this holds are exactly those such that pt(L) is a T D -space, as shown in [14]. By looking at the diagram, we see that these are exactly the frames for which there is a perfect correspondence between the spatial sublocales of L and the subspaces of pt(L).…”

“…In [14], the relation between sublocales of a frame L and subspaces of its spectrum pt(L) is explored. Spatial sublocales and sober subspaces are seen as fixpoints of the following adjunction of posets.…”

“…In [14] the matter of the relation between subspaces and sublocales is explored. Recall that there it is proven that we have an adjunction…”

“…We wish to stress that the behaviour of the operator pt D • M: P(pt D (L)) → P(pt D (L)) exhibits a difference between the spatial-sober duality and the T D -duality. In the classical duality (see [14]) we have an adjunction…”

“…The properties of this sublocale have been studied in [1]. We also have sp[S(L)] (see [14]), the sublocale of S(L) consisting of those sublocales which are spatial -that is, those sublocales which are the frame of opens of some topological space.…”