Completeness types for uniformity theory on textures

Yıldız, Filiz

doi:10.2298/fil1501159y

Cited by 4 publications

(2 citation statements)

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“…Recent works on textures show that they are also useful model for rough set theory [8] and semi-separation axioms [10]. On the other hand, it was given various types of completeness for diuniform texture spaces [13]. As an expanded of classical metric spaces, the dimetric notion on texture spaces was firstly defined in [11].…”

Section: Introductionmentioning

confidence: 99%

Metric spaces and textures

Dost

2017

Appl. Gen. Topol.

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<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>Textures are point-set setting for fuzzy sets, and they provide a framework for the complement-free mathematical concepts. Further dimetric on textures is a gener- alization of classical metric spaces. The aim of this paper is to give some properties of dimetric texture space by using categorical approach. We prove that the category of clas- sical metric spaces is isomorphic to a full subcategory of dimetric texture spaces, and give a natural transformation from metric topologies to dimetric ditopologies. Further, it is pre- sented a relation between dimetric texture spaces and quasi-pseudo metric spaces in the sense of J. F. Kelly. </span></p></div></div></div>

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Section: Introductionmentioning

confidence: 99%

Metric spaces and textures

Dost

2017

Appl. Gen. Topol.

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“…Some of the links with Hutton spaces and fuzzy topologies are expressed in a categorical setting in [14]. In addition, there are close and deep relationships between the bitopological and ditopological spaces as shown in [11,12] and [15,16]. In this study, we will use those close relationships insofar as the inverse systems and their inverse limits are concerned in a categorical view.…”

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confidence: 99%

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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