Inverse systems and limits in the category of ditopological plain spaces

Yıldız, Filiz

doi:10.1016/j.topol.2017.05.005

Cited by 1 publication

(17 citation statements)

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“…As it is stated before, in [2,3,4] we have a few methods, such as product space, subtexture space and quotient space, to derive a new ditopological space from two or more ditopological spaces just like classical case. Recently, it is seen in [17,18] that the another method used to construct a new ditopological space is the theory of ditopological inverse systems and their limit spaces under the name ditopological inverse limits as the subspaces of ditopological product spaces described in [3,4,18].…”

Section: Introductionmentioning

confidence: 99%

“…Later, in [18], the theory of inverse systems and inverse limits is handled first-time in the ditopological textural context and we gave a detailed analysis of the theory of ditopological inverse systems and inverse limits insofar as the category ifPDitop whose objects are the ditopological texture spaces which have plain texturing and morphisms are the bicontinuous, w-preserving point functions, is concerned. (For a detailed information and some basic facts about the point-functions between texture spaces, see [3,10,11]).…”

Section: Introductionmentioning

confidence: 99%

“…Especially, this paper will present some intriguing connections between the bitopological inverse systems -limit spaces and their ditopological counterparts, in a categorical -functorial setting. Here we will continue to work within the same framework given in [17,18] that are the major sources of the topic on which we study.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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].…”

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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: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…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].…”

mentioning

confidence: 99%

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