Proximity structure on generalized topological spaces

Mukherjee, M.; Mandal, Debabrata; Dey, Dipankar

doi:10.1007/s13370-018-0629-6

Cited by 5 publications

(16 citation statements)

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“…A nonempty family H of subsets of X is called hereditary class if it is closed under subsets. e set of all hereditary classes on X is denoted by H. Definition 2 (see [26]). A binary relation δ μ on the power set P(X) of a set X is called a μ-proximity (μ-nearness) on X and (X, δ μ ) is a μ-proximity (μ-nearness) space if, for all G, K⊆X, δ μ satisfies the following axioms: (2), and (3).…”

Section: Preliminariesmentioning

confidence: 99%

“…We denote by ℘(X) the family of all basic μ-proximities on X. Henceforth, we write xδ μ G for x { }δ μ G. Several properties of the relation δ μ on X have been mentioned with details in [26].…”

Section: Preliminariesmentioning

confidence: 99%

“…Many researchers have worked with weaker axioms than those of the fundamental concept ofEfremovic proximity space [25] enabling them to introduce an arbitrary topology on the underlying set with nice properties, and the theory possesses deep results, rich machinery, and tools. In 2019, Mukherjee et al [26] constructed a generalized proximity structure, named μ-proximity on set X, which induces a generalized topology(GT) on X. Also, Yildirim [27] constructed a generalized μ-proximity structure by using hereditary class on a set.…”

Section: Introductionmentioning

confidence: 99%

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[Retracted] New Types of μ‐Proximity Spaces and Their Applications

Hosny

Al-shami

Mhemdi

2022

Journal of Mathematics

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Near set theory supplies a major basis for the perception, differentiation, and classification of elements in classes that depend on their closeness, either spatially or descriptively. This study aims to introduce a lot of concepts; one of them is μ -clusters as the useful notion in the study of μ -proximity (or μ -nearness) spaces which recognize some of its features. Also, other types of μ -proximity, termed R μ -proximity and O μ -proximity, on X are defined. In a μ -proximity space X , δ μ , for any subset K of X , one can find out nonempty collections δ μ K = G ⊆ X ∣ K δ ¯ μ G , which are hereditary classes on X . Currently, descriptive near sets were presented as a tool of solving classification and pattern recognition problems emerging from disjoint sets; hence, a new approach to basic μ -proximity structures, which depend on the realization of the structures in the theory of hereditary classes, is introduced. Also, regarding to specific options of hereditary class operators, various kinds of μ -proximities can be distinguished.

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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[Retracted] New Types of μ‐Proximity Spaces and Their Applications

Hosny

Al-shami

Mhemdi

2022

Journal of Mathematics

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“…In Section 3, we also prove that a μ-uniform space satisfies a sort of complete regularity condition. Finally in Section 4, we establish that for a μ-uniform space, there exists a μ-proximity relation [2] such that the same generalized topology originates from both the structures.…”

Section: Introduction and Prerequisitesmentioning

confidence: 99%

“…[2] A binary relation δ μ on the power set PðX Þ of a set X is called a μ-proximity on X if δ μ satisfies the following axioms: Aδ μ B iff Bδ μ A; ∀A; B ∈ PðXÞ(2) If Aδ μ B, A ⊆ C and B ⊆ D, then Cδ μ D…”

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confidence: 99%

Uniformity on generalized topological spaces

Dey¹,

Mandal

Mukherjee

2021

AJMS

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PurposeThe present article deals with the initiation and study of a uniformity like notion, captioned μ-uniformity, in the context of a generalized topological space.Design/methodology/approachThe existence of uniformity for a completely regular topological space is well-known, and the interrelation of this structure with a proximity is also well-studied. Using this idea, a structure on generalized topological space has been developed, to establish the same type of compatibility in the corresponding frameworks.FindingsIt is proved, among other things, that a μ-uniformity on a non-empty set X always induces a generalized topology on X, which is μ-completely regular too. In the last theorem of the paper, the authors develop a relation between μ-proximity and μ-uniformity by showing that every μ-uniformity generates a μ-proximity, both giving the same generalized topology on the underlying set.Originality/valueIt is an original work influenced by the previous works that have been done on generalized topological spaces. A kind of generalization has been done in this article, that has produced an intermediate structure to the already known generalized topological spaces.

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