Scatteredness: Joins of Closed Sublocales

Picado, Jorge; Pultr, Aleš

doi:10.1007/978-3-030-53479-0_9

Cited by 1 publication

(3 citation statements)

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“…endowed with the inclusion order inherited from S(L). In the recent paper [10], Picado, Pultr and Tozzi show (a.o.) that S c (L) is always a frame which is embedded as a joinsublattice in the coframe S(L).…”

Section: Joins Of Closed (Resp Complemented) Sublocalesmentioning

confidence: 95%

“…that S c (L) is always a frame which is embedded as a joinsublattice in the coframe S(L). One of the main results from [10] is that if L is subfit, and only in that case, S c (L) is a Boolean algebra and coincides precisely with the Booleanization of S(L).…”

Section: Joins Of Closed (Resp Complemented) Sublocalesmentioning

confidence: 99%

“…Recently, some of these special sublocales and subsets have enjoyed special attention. We have the system of all joins of closed sublocales S c (L) (see [10]), which coincides with the smallest dense sublocale of S(L) if and only if the frame L satisfies a weak separation axiom called subfitness. We then have S b (L), the smallest dense sublocale of S(L).…”

Section: Introductionmentioning

confidence: 99%

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

Arrieta,

Suarez

2020

Preprint

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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: 95%

Section: Joins Of Closed (Resp Complemented) Sublocalesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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