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

Igor Arrieta; Anna Laura Suarez

doi:10.1016/j.topol.2021.107614

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

Igor Arrieta^*, Anna Laura Suarez

^*Corresponding author for this work

Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

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).

Original language	English
Article number	107614
Number of pages	22
Journal	Topology and its Applications
Volume	291
Early online date	29 Jan 2021
DOIs	https://doi.org/10.1016/j.topol.2021.107614
Publication status	Published - 15 Mar 2021

Keywords

Locale
TD -space
Coframe
Totally spatial frame

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title = "The coframe of D-sublocales of a locale and the TD-duality",

abstract = "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). ",

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T1 - The coframe of D-sublocales of a locale and the TD-duality

AU - Arrieta, Igor

AU - Suarez, Anna Laura

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N2 - 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).

AB - 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).

KW - Locale

KW - TD -space

KW - Coframe

KW - Totally spatial frame

U2 - 10.1016/j.topol.2021.107614

DO - 10.1016/j.topol.2021.107614

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JO - Topology and its Applications

JF - Topology and its Applications

M1 - 107614

ER -

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

Abstract

Keywords

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