Characterising points which make P-frames

Dube, Themba; Ighedo, Oghenetega

doi:10.1016/j.topol.2015.12.017

Cited by 3 publications

(3 citation statements)

References 12 publications

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“…, is an example of a functorial neighbourhood system on X (Theorem 3.38 & Definition 4.3 [49]). T -neighbourhood systems have been used extensively in [45,46]. Since for a localic map X f − → Y , the preimage f −1 does not preserve arbitrary joins, with X and Y empowered with T -neighbourhood systems, f is merely a preneighbourhood morphism and not a neighbourhood morphism.…”

Section: The Forgetful Functor Pnbd[a]mentioning

confidence: 99%

“…Notable amidst them are closures with respect to functorial neighbourhood systems (see Theorem 3.38 and Definition 4.3, [49]) on locales, groups and commutative rings without identity. On locales it is known from [49] that the T -neighbourhood systems (see equation (2.19), [45,46]) are functorial; it is shown here that the closure with respect to the T -neighbourhood system is precisely the usual closure of a sublocale (see §III.8, [58]); furthermore every localic map X f − → Y is continuous with respect to any preneighbourhood system µ on X, ϕ on Y if µ is larger than the T -neighbourhood system on X and ϕ is smaller than the T -neighbourhood system on Y (see §3.5). Example 2.21 shows the normal closure induces a functorial neighbourhood system ν X on each group X.…”

Section: Introductionmentioning

confidence: 99%

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Internal Neighbourhood Structures II: Closure and closed morphisms

Ghosh

2023

CGASA

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Internal preneighbourhood spaces inside any finitely complete category with finite coproducts and proper factorisation structure were first introduced in [49]. This paper proposes a closure operation on internal preneighbourhood spaces and investigates closed morphisms and its close allies. Consequently it introduces analogues of several well-known classes of topological spaces for preneighbourhood spaces. Some preliminary properties of these spaces are established in this paper. The results of this paper exhibit that preneighbourhood systems are more general than closure operators and conveniently allows identifying properties of classes of morphisms independent of continuity of morphisms with respect to induced closure operators.

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Section: The Forgetful Functor Pnbd[a]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Internal Neighbourhood Structures II: Closure and closed morphisms

Ghosh

2023

CGASA

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“…In the case of the context (Loc, Epi, RegMon), the functorial neighbourhood system utilised in [5,6], henceforth referred to as T -neighbourhood system:…”

Section: Definitionmentioning

confidence: 99%

Internal Neighbourhood Structures III: Finite Sum of Subobjects

Ghosh

2020

Preprint

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This paper develops further on internal preneighbourhood spaces [see 7, 8] in presenting the notion of internal Hausdorff spaces and the Hausdorff reflection. The development depends on the notions of proper morphisms, separated morphisms and separated objects, which are also developed in this paper. The Hausdorff reflection is described in three different ways: firstly as the largest subobject of the binary product whose components are indistinguishable by any internal Hausdorff space valued preneighbourhood morphism, secondly as the smallest effective equivalence relation whose quotient is an internal Hausdorff space and thirdly in admissibly well powered categories by transfinite induction on quotients by the diagonal.

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Internal neighbourhood structures

Ghosh

2020

Algebra Univers.

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The main aim of this paper is to provide a description of neighbourhood operators in finitely complete categories with finite coproducts and a proper factorisation system such that the semilattice of admissible subobjects make a distributive complete lattice. The equivalence between neighbourhoods, Kuratowski interior operators and pseudo-frame sets is proved. Furthermore the categories of internal neighbourhoods is shown to be topological. Regular epimorphisms of categories of neighbourhoods are described and conditions ensuring hereditary regular epimorphisms are probed. It is shown the category of internal neighbourhoods of topological spaces is the category of bitopological spaces, while in the category of locales every locale comes equipped with a natural internal topology.

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Characterising points which make P-frames

Cited by 3 publications

References 12 publications

Internal Neighbourhood Structures II: Closure and closed morphisms

Internal Neighbourhood Structures II: Closure and closed morphisms

Internal Neighbourhood Structures III: Finite Sum of Subobjects

Internal neighbourhood structures

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