Extended real dicompactness and an application to Hutton spaces

Yıldız, Filiz; Brown, L. M.

doi:10.1016/j.fss.2013.03.012

Cited by 9 publications

(4 citation statements)

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“…As shown in [22] Moreover, P (r,0) = Ur, Q (r,0) = Vr; P (r,1) = Q (r,1) = Vr; P∞ = Q∞ = M R . As noted in [22] this texture is not nearly plain since ∞ is a non-plain point whose q-set is not equal to the q-set of any of the plain points. If we take τ = {Vr | r ∈ R} ∪ {M R , ∅}, κ = {M R , ∅} then much as in the above example we see that τ * = κ * = {M R , ∅}, whence (M R , M R , τ, κ) is nearly dicompact.…”

Section: (Iii) Straightforward From (I) and (Ii)mentioning

confidence: 86%

“…Consider the Hutton texture of the real texture (R, R) given in Examples 1.1 (4). As shown in [22] Moreover, P (r,0) = Ur, Q (r,0) = Vr; P (r,1) = Q (r,1) = Vr; P∞ = Q∞ = M R . As noted in [22] this texture is not nearly plain since ∞ is a non-plain point whose q-set is not equal to the q-set of any of the plain points.…”

Section: (Iii) Straightforward From (I) and (Ii)mentioning

confidence: 99%

“…Textures and ditopological texture spaces were first introduced by the first author as a point-based setting for the study of fuzzy sets, and work continues in this direction, see for example [1,2,3,4,5], and the recent work of Tiryaki [18], Özçağ and Brown [15], Yıldız and Brown [22]. On the other hand, textures provide a very convenient setting for the investigation of complement-free concepts in general, so much of the recent work does not involve fuzzy sets explicitly.…”

Section: Introductionmentioning

confidence: 99%

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Near Compactness of Ditopological Texture Spaces

Brown

2015

HJMS

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The authors consider the notions of near compactness, near cocompactness, near stability, near costability and near dicompactness in the setting of ditopological texture spaces. In particular preservation of these properties under surjective R-dimaps, co-R-dimaps and bi-R-dimaps is investigated and non-trivial characterizations of near dicompactness are given which generalize those for dicompactness. The notions of semiregularization, semicoregularization and semibiregularization are defined and used to give generalizations of Mrówka's Theorem for near compactness and near cocompactness, and of Tychonoff's Theorem for near compactness, near cocompactness and near dicompactness. Also, results related to pseudo-open and pseudo-closed sets are presented. Finally, examples are given of co-T1 nearly dicompact ditopologies on textures which are not nearly plain and an open question is posed. 2000 AMS Classification: 54 A 05, 54 C 10.

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Section: (Iii) Straightforward From (I) and (Ii)mentioning

confidence: 86%

Section: (Iii) Straightforward From (I) and (Ii)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Near Compactness of Ditopological Texture Spaces

Brown

2015

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“…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. 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].…”

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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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Textures

Diker

2023

Studies in Fuzziness and Soft Computing

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Extended real dicompactness and an application to Hutton spaces

Cited by 9 publications

References 21 publications

Near Compactness of Ditopological Texture Spaces

Near Compactness of Ditopological Texture Spaces

Some categorical aspects of the inverse limits in ditopological context

Textures

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