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

摘要

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) congruent to 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). (c) 2021 Elsevier B.V. All rights reserved.