Abstract
In this article, we will define the new notions (e.g., b − θ -neighborhood system of point, b − θ -closure (interior) of a set, and b − θ -closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of the above five notions in detail. Several basic results (for instance, Definition 7, Theorem 3 (iii), (v), and (vi), Theorem 5, Theorem 9, and Theorem 4.6) in classical topology are generalized in fuzzy logic. In addition to, we will show that every fuzzifying b − θ -closed set is fuzzifying γ -closed set (by Theorem 3 (vi)). Further, we will study the notion of fuzzifying b − θ -derived set and fuzzifying b − θ -boundary set and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly b − θ -continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly b − θ -continuous mapping, fuzzifying strongly b − θ -irresolute mapping, and fuzzifying weakly b − θ -irresolute mapping along with different conditions for their existence are obtained.
Highlights
Introduction and PreliminariesIn classical topology, the notions of b-open set, b-closed set, and strongly θ − b-continuous mapping are presented in [1, 2]
Benchalli and Karnel [5] presented a novel form of fuzzy subset named fuzzy b-open set, and some basic properties are proved and their relations with different fuzzy sets in fuzzy topological spaces are investigated
In 2017, Dutta and Tripathy [6] introduced a new kind of open set named fuzzy b − θ open set
Summary
Introduction and PreliminariesIn classical topology, the notions of b-open set, b-closed set, and strongly θ − b-continuous mapping are presented in [1, 2]. We present the basic notions related to fuzzifying topological space as follows. Τ (i.e., τ ∈ F (2X), 2X the set of all subsets of a set X) is called a fuzzifying topological space (for short, FTS(X, τ)) if we have the following three conditions: (i) τ(X) τ(∅) 1 (ii) ∀Φ, Ψ, τ(Φ⊓Ψ) ≥ τ(Φ)∧τ(Ψ) (iii) ∀Φλ: λ ∈ Λ, τ(⊔λ∈ΛΦλ) ≥ infλ∈Λτ(Φλ) E several notions of FTS (X, τ) are given as follows (∀Φ, Ψ ∈ 2X): (i) F (i.e., F ∈ F (2X)) is called the set of all fuzzifying closed sets if Φ ∈ F ≔ Φc ∈ τ, where Φc X − Φ is the complement of Φ
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