Some new sets and a new decomposition of fuzzycontinuity, fuzzy almost strongI

In this paper, we discuss the notion of fuzzy strong β I -open sets. We introduce fuzzy almost I almost strong I -open sets, fuzzy SβI -set, fuzzy DI sets closely related with fuzzy strong β Additionally, we investigate some characterizations and properties of these sets. With the help of fuzzy


Introduction and Preliminaries
Fuzziness is one of the most important and useful concepts in the modern scientific studies. In 1965, Zadeh [11] first introduced the notion of fuzzy sets. In 1945, Vaidyanathaswamy [9] introduced the concepts of ideal topological spaces. In 1990, Jankovic and Hamlett [4] have defined the concept of I in ideal topological space. In 1997, Mahmoud [6] and sarkar [8] independently presented some of the ideal concepts in the fuzzy trend and studied many other properties. Decomposition of many problems in the fuzzy topology. It becomes very interesting when decomposition is done via fuzzytopological ideals.
Throughout this paper, X represents a nonempty fuzzy set and fuzzy subset A of X , characterized by a membership function in the sense of Zadeh [11]. The basic fuzzy sets are the empty set, the whole set and the class of all fuzzy subsets of X which will be denoted by 0, 1 and I subfamily τ of I X will denote topology of fuzzy sets on I topological space in Chang's sense. A fuzzy point in X with support x x α . For a fuzzy subset A of X , Cl(A), Int(A) and 1 interior and fuzzy complement of A. A nonempty collection I of fuzzy subsets of X is called a fuzzyideal [8]  Fuzziness is one of the most important and useful concepts in the modern scientific studies. In 1965, Zadeh [11] first introduced the notion of fuzzy sets. In 1945, Vaidyanathaswamy [9] introduced the concepts of ideal nd Hamlett [4] have defined the concept of I -open set via localfunction in ideal topological space. In 1997, Mahmoud [6] and sarkar [8] independently presented some of the ideal concepts in the fuzzy trend and studied many other properties. Decomposition of fuzzy continuity is one of the many problems in the fuzzy topology. It becomes very interesting when decomposition is done via Throughout this paper, X represents a nonempty fuzzy set and fuzzy subset A of X , characterized by a membership function in the sense of Zadeh [11]. The basic fuzzy sets are the empty set, the whole set and the class of all fuzzy subsets of X which will be denoted by 0, 1 and I ill denote topology of fuzzy sets on I X as defined by Chang [3]. By (X,τ),we mean a fuzzy topological space in Chang's sense. A fuzzy point in X with support x ∈ X and value α(0 < α ≤ 1)is denoted by . For a fuzzy subset A of X , Cl(A), Int(A) and 1 -Awill respectively, denote the fuzzy closure, fuzzy interior and fuzzy complement of A. A nonempty collection I of fuzzy subsets of X is called a fuzzyideal [8]

Fuzzy topological ideal; fuzzy strong β -I -SβI -set; fuzzy almost Iopen set
Fuzziness is one of the most important and useful concepts in the modern scientific studies. In 1965, Zadeh [11] first introduced the notion of fuzzy sets. In 1945, Vaidyanathaswamy [9] introduced the concepts of ideal open set via localfunction in ideal topological space. In 1997, Mahmoud [6] and sarkar [8] independently presented some of the ideal fuzzy continuity is one of the many problems in the fuzzy topology. It becomes very interesting when decomposition is done via Throughout this paper, X represents a nonempty fuzzy set and fuzzy subset A of X , denoted by A ≤ X , is characterized by a membership function in the sense of Zadeh [11]. The basic fuzzy sets are the empty set, the whole set and the class of all fuzzy subsets of X which will be denoted by 0, 1 and I X respectively. A as defined by Chang [3]. By (X,τ),we mean a fuzzy X and value α(0 < α ≤ 1)is denoted by ill respectively, denote the fuzzy closure, fuzzy interior and fuzzy complement of A. A nonempty collection I of fuzzy subsets of X is called a fuzzyideal [8] if International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 @ IJTSRD | Available Online @ www.ijtsrd.com A fuzzy ideal topological space, denoted by (X,τ,I)means a fuzzy topological space with a fuzzy ideal I and fuzzy topology τ . For (X,τ,I), the fuzzy local function of A ≤ X with respect to τ and I is denoted by A * (τ,I)(briefly A * )and is defined as A * (τ,I) = of the fuzzy points x such that if U ∈ τ(x) and E E(y). Fuzzy closure operator of a fuzzy set in(X,τ,I)is defined as τ * (I)means an extension of fuzzy topological space than τ via fuzzy ideal which is constructed by considering the class β = {U − E : U ∈ τ,∈I}as a base [8]. This topology of fuzzy sets is considered as generalizati one.
First, we shall recall some definitions used in the sequel. Lemma 1.1. [8] Let(X,τ,I) be fuzzy ideal topological space and A, B subsets of X.
The following properties hold: Let(X,τ)be an ideal topological space with an arbitrary index ρ(X)the power set of X. If {A α : α ∈ ∆}≤ A fuzzy ideal topological space, denoted by (X,τ,I)means a fuzzy topological space with a fuzzy ideal I and fuzzy topology τ . For (X,τ,I), the fuzzy local function of A ≤ X with respect to τ and I is denoted by (τ,I) = ∨{x ∈ X : A ∧ U / I for every U ∈ τ(x)}. While A τ(x) and E ∈I ,then there is at least one y ∈ X for which U(y) + A(y) E(y). Fuzzy closure operator of a fuzzy set in(X,τ,I)is defined as Cl * (A) = A ∨ A * . In(X,τ,I),the collection (I)means an extension of fuzzy topological space than τ via fuzzy ideal which is constructed by considering I}as a base [8]. This topology of fuzzy sets is considered as generalizati First, we shall recall some definitions used in the sequel. [8] Let(X,τ,I) be fuzzy ideal topological space and A, B subsets of X.
[5] Let(X,τ)be an ideal topological space with an arbitrary index ∆, I an ideal of subsets of X and ≤ ρ(X), then the following property holds: A subset A of a space(X,τ,I)is said to be ≤ Int(Cl * (Int(A))), open [10] if A ≤ Cl(Int(Cl * (A))).
[10] A subset A of a space(X,τ)is said to be fuzzy β -open if [12] A A subset A of a fuzzy ideal topological space(X,τ,I)is said to be almost fuzzy strong )). We denote the family of all fuzzy almost strong-I -open set of (X,τ,I) by FasIO(X). Page: 1611 A fuzzy ideal topological space, denoted by (X,τ,I)means a fuzzy topological space with a fuzzy ideal I and fuzzy topology τ . For (X,τ,I), the fuzzy local function of A ≤ X with respect to τ and I is denoted by τ(x)}. While A * is the union X for which U(y) + A(y) − 1 > . In(X,τ,I),the collection (I)means an extension of fuzzy topological space than τ via fuzzy ideal which is constructed by considering I}as a base [8]. This topology of fuzzy sets is considered as generalization of the ordinary ∆, I an ideal of subsets of X and dense in itself if A ≤ A * . This shows that A ∈ FsβIO(X).