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

Igor Arrieta*, Anna Laura Suarez

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The notion of D-sublocale is explored. This is the notion analogue to that of sublocale in the duality of TD-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 SD(L)  of the coframe S(L) of all its sublocales. It is also shown that the spatialization spD[SD(L)] of SD(L) consists precisely of those D-sublocales of L which are TD-spatial. Additionally, frames such that we have SD(L) ≅ P(ptD(L)))  — that is, those such that D-sublocales perfectly represent subspaces — are characterized as those TD-spatial frames such that SD(L)is the Booleanization of S(L).
Original languageEnglish
Article number107614
Number of pages22
JournalTopology and its Applications
Early online date29 Jan 2021
Publication statusPublished - 15 Mar 2021


  • Locale
  • TD -space
  • Coframe
  • Totally spatial frame


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