The coframe of D-sublocales of a locale and the TD-duality

The notion of 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 S_D(L)  of the coframe S(L) of all its sublocales. It is also shown that the spatialization sp_D[S_D(L)] of S_D(L) consists precisely of those D-sublocales of L which are T_D-spatial. Additionally, frames such that we have S_D(L) ≅ P(pt_D(L)))  — that is, those such that D-sublocales perfectly represent subspaces — are characterized as those T_D-spatial frames such that S_D(L)is the Booleanization of S(L).