On joins of complemented sublocales

Arrieta, Igor

doi:10.1007/s00012-021-00757-y

Cited by 3 publications

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“…The latter is precisely the system consisting smooth sublocales, that is, those which are joins of complemented sublocales (or equivalently joins of locally closed sublocales). A study of the properties of S b (L) may be found in [1].…”

Section: Joins Of Closed (Resp Complemented) Sublocalesmentioning

confidence: 99%

“…We then have S b (L), the smallest dense sublocale of S(L). The properties of this sublocale have been studied in [1]. We also have sp[S(L)] (see [14]), the sublocale of S(L) consisting of those sublocales which are spatial -that is, those sublocales which are the frame of opens of some topological space.…”

Section: Introductionmentioning

confidence: 99%

“…But since p is covered, by Lemma 3.16 this implies that it is also an absolutely essential prime of a. (8) =⇒(1). Suppose that L is spatial, and that every element other than 1 has a covered absolutely essential prime above it.…”

mentioning

confidence: 97%

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

Arrieta,

Suarez

2020

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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).

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Section: Joins Of Closed (Resp Complemented) Sublocalesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%