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

TL;DR

This paper investigates D-sublocales within frames, establishing their properties, their relation to $T_D$-spaces, and characterizing frames where D-sublocales perfectly represent subspaces, thus advancing the duality theory in locale theory.

Contribution

It introduces the notion of D-sublocales, analyzes their structure, and characterizes frames where D-sublocales correspond exactly to subspaces, extending the duality between locales and $T_D$-spaces.

Findings

D-sublocales form a dense sublocale of all sublocales.

Spatialization of D-sublocales corresponds to $T_D$-spatial D-sublocales.

Frames with D-sublocales as Booleanization represent subspaces precisely.

Abstract

The notion of \emph{D-sublocale} is explored. This is the notion analogue to that of sublocale in the duality of $T_D$spaces. A sublocale $S$ of a frame $L$ is a D-sublocale if and only if the corresponding localic map preserves the property of being a covered prime. It is shown that for a frame $L$ the system of those sublocales which are also D-sublocales form a dense sublocale $\mathsf{S}_D(L)$ of the coframe $\mathsf{S}(L)$ of all its sublocales. It is also shown that the spatialization $\mathsf{sp}_D[\mathsf{S}_D(L)]$ of $\mathsf{S}_D(L)$ consists precisely of those D-sublocales of $L$ which are $T_D$-spatial. Additionally, frames such that we have $\mathsf{S}_D(L)\cong \mathcal{P}(\mathsf{pt}_D(L))$ -- that is, those such that D-sublocales perfectly represent subspaces -- are characterized as those $T_D$-spatial frames such that $\mathsf{S}_D(L)$ is the Booleanization of \mathsf{S}(L).