Plain ditopological texture spaces

Tiryaki, İsmail U.; Brown, L. M.

doi:10.1016/j.topol.2011.06.051

Cited by 10 publications

(11 citation statements)

References 17 publications

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“…Clearly, it defines a functor H : Bitop w0 → ifPDitop 0 as mentioned in [9]. Note that this concrete functor is a variant of the functor with the same name considered in [12,15] in connection with real dicompactness.…”

Section: Relationships Between the Inverse Systems-limits In The Catementioning

confidence: 99%

“…It may be verified that H preserves the other basic ditopological separation axioms, besides T 0 axiom. Consequently, we have the following fact from [9,12]:…”

Section: F Yıldızmentioning

confidence: 99%

“…In addition, a ditopological texture space or shortly ditopological space with respect to a ditopology (τ, κ) on the texture (S, S) is denoted by (S, S, τ, κ).There is now a considerable literature on the theory of ditopological spaces. An adequate introduction to this theory and the motivation for its study may be obtained from [2,3,4,8,9,10,13]. As will be clear from these general references, it is shown that ditopological spaces provide a unified setting for the study of topology, bitopology and fuzzy topology on Hutton algebras.…”

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confidence: 99%

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Some categorical aspects of the inverse limits in ditopological context

Yıldız

2018

Appl. Gen. Topol.

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This paper considers some various categorical aspects of the inverse systems (projective spectrums) and inverse limits described in the category ifPDitop, whose objects are ditopological plain texture spaces and morphisms are bicontinuous point functions satisfying a compatibility condition between those spaces. In this context, the category Inv ifPDitop consisting of the inverse systems constructed by the objects and morphisms of ifPDitop, besides the inverse systems of mappings, described between inverse systems, is introduced, and the related ideas are studied in a categorical -functorial setting. In conclusion, an identity natural transformation is obtained in the context of inverse systems -limits constructed in ifPDitop and the ditopological infinite products are characterized by the finite products via inverse limits.The origins of the study of inverse limits date back to the 1920 's. Classical theory of inverse systems and inverse limits are important in the extension of homology and cohomology theory. An exhaustive discussion of inverse systems which are in the some classical categories such as Set, Top, Grp and Rng defined in [1], was presented by the paper [5] which is a milestone in the development of that theory.As is the case with products, the inverse limit might not exist in any category in general whereas inverse systems exist in every category. Note from that [5] inverse limits exist in any category when that category has products of objects and the equalizers [1] of pairs of morphisms, in other words, the inverse limits exist in any category if the category is complete, in the sense of [1]. Additionally, an inverse system has at most one limit. That is, if an inverse limit of any inverse system exists in any category C, this limit is unique up to C-isomorphism. Incidentally, inverse limits always exist in the categories Set, Top, Grp and Rng. Note also that inverse limits are generally restricted to diagrams over directed sets.Similarly, a suitable theory of inverse systems and inverse limits for the categories consisting of textures and ditopological spaces is handled first-time in [17] and [18].Incidentally, let 's recall the notions of texture and ditopology introduced in 1993, by Lawrence M. Brown : For a nonempty set S, the family S ⊆ P(S) is called a texturing on S if (S, ⊆) is a point-separating, complete, completely distributive lattice containing S and ∅, with meet coinciding with intersection and finite joins with union. The pair (S, S) is then called a texture. If S is closed under arbitrary unions, it is called plain texturing and (S, S) is called plain texture. Since a texturing S need not be closed under the operation of taking the set-complement, the notion of topology is replaced by that of dichotomous topology or ditopology, namely a pair (τ, κ) of subsets of S, where the set of open sets τ and the set of closed sets κ, satisfy the some dual conditions. Hence a ditopology is essentially a "topology" for which there is no a priori relation between the open and closed sets. In ad...

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Section: Relationships Between the Inverse Systems-limits In The Catementioning

confidence: 99%

“…It may be verified that H preserves the other basic ditopological separation axioms, besides T 0 axiom. Consequently, we have the following fact from [9,12]:…”

Section: F Yıldızmentioning

confidence: 99%

mentioning

confidence: 99%

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Some categorical aspects of the inverse limits in ditopological context

Yıldız

2018

Appl. Gen. Topol.

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“…An adequate introduction to this theory and the motivation for its study may be obtained from [2,3,4,8,9,10,13]. As will be clear from these general references, it is shown that ditopological spaces provide a unified setting for the study of topology, bitopology and fuzzy topology on Hutton algebras.…”

Section: Introductionmentioning

confidence: 99%

Inverse systems and limits in the category of ditopological plain spaces

Yıldız

2017

Topology and its Applications

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“…A complementation σ on a texture (S, S) is called "grounded" [14] if there is an involution s → s on S such that σ(P s ) = Q s and σ(Q s ) = P s (s will be denoted by σ(s)) for all s ∈ S and in this case the complemented texture space (S, S, σ) is called "complemented grounded texture space". It is obtained that a complemented plain texture is grounded in [19].…”

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confidence: 99%