Abstract
This paper aims to present the notion of (i, j)*-fuzzy γ-open set in a fuzzy bitopological space as a parallel form of (i, j)-fuzzy γ-open set due to Tripathy and Debnath (2013) [17] and show that both of them are independent concepts. Then we extend our study to (i, j)*-generalized fuzzy γ-closed set and (i, j)*-γ-generalized fuzzy closed set. We show that (i, j)*-γ-generalized fuzzy closed set and (i, j)*-generalized fuzzy γ-closed set are also independent of each other in nature. Though every (i, j)*-fuzzy γ-closed set is a (i, j)*-generalized fuzzy γ-closed set but with (i, j)*-γ-generalized fuzzy closed set, the same relation is not linear. Similarly though every (i, j)*-fuzzy closed set is (i, j)*-γ-generalized fuzzy closed set but it is independent to (i, j)*-generalized fuzzy γ-closed set. Various properties related to (i, j)*-generalized fuzzy γ-closed set are also studied. Finally, (i, j)*-generalized fuzzy γ-continuous function and (i, j)*-generalized fuzzy γ-irresolute functions are introduced and interrelationships among them are established. We characterized these functions in different directions for different applications.
Highlights
Levine (1970) first initiated the concept of generalized closed set in a topological space
Bitopological space was first introduced by Kelly (1963) and till various constructive works have been going on in this particular field viz. Tripathy and Acharjee (2014), Tripathy and Sarma (2011, 2012, 2013, 2014), Tripathy and Debnath (2015, 2019)
No report has been found till date, in showing the interest for extending this study in fuzzy bitopological space, which motivated us to define the concept of (i, j)∗-generalized fuzzy closed set and (i, j)∗-generalized fuzzy γ-closed set therein
Summary
Levine (1970) first initiated the concept of generalized closed set in a topological space. Bin Sahana (1991), as well as Singal and Prakash (1991) initiated the notion of fuzzy pre-open set in a fuzzy topological space and using this set Bhattacharya (2017) defined fuzzy γ∗-open set therein. Following these definitions, here we introduce (i, j)∗-fuzzy pre-open set and (i, j)∗-fuzzy γ-open set in a fbts. A fuzzy subset μ in a fbts (X, τi, τj) is called a (i, j)-fuzzy generalized closed set if τj-cl(μ) ≤ η, whenever μ ≤ η and η ∈ τi-F O(X), where τi-F O(X) is the family of all τi fuzzy open sets.
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