On fuzzy b -θ open sets in fuzzy topological space

1 Introduction

The study of fuzzy set was initiated with the famous paper of Zadeh [10] in 1965. Since then the notion of fuzziness has been applied in all the branches of science and technology. It has been applied in mathematical analysis for introducing and investigating different classes of sequence spaces of fuzzy numbers by Tripathy and Baruah [6], Tripathy and Sarma [9] and many others in the recent past.

Chang [3] in 1968 introduced the notion of fuzzy topological space and opened the way for the subsequent growth of the numerous fuzzy topological concepts. Mixed fuzzy topological space was studied by Tripathy and Ray [8]. As a generalization of open sets, b-open sets where introduce and studied by Andrijevic [1]. In this paper we introduce the concept of fuzzy b-θ open set in fuzzy topological space. Tripathy and Debnath [7] studied about the notion of b-open fuzzy topological spaces.

2 Preliminaries

Let X be a non-empty set and I, the unit interval [0, 1]. A fuzzy set A in X has been characterized by a function μ _A  : X → I, where μ _A is called the membership function of A. μ _A  (x) represents the membership grade of x in A. The empty fuzzy set is defined by μ _φ  (x) =0 for all x ∈ X, denoted by 0 _X . Also X can be regarded as a fuzzy set in itself defined by μ _X  (t) =1 for all t ∈ X, denoted by 1 _X . Further, an ordinary subset A of X can also be regarded as a fuzzy set in X if its membership function is taken as usual characteristic function of A that is μ _A  (x) =1, for all x ∈ X and μ _A  (x) =0 for all x ∈ X - A. Two fuzzy sets A and B are said to be equal if μ _A  (x) = μ _B  (x) for all x ∈ X. A fuzzy set A is said to be contained in a fuzzy set B, written as A ⊆ B, if μ _A  ≤ μ _B . Complement of a fuzzy set A in X is a fuzzy set A in X defined by μ _{A ^c}  = 1 - μ _A . Union and intersection of a collection {A _i  : i ∈ I} of fuzzy sets in X, to be written as V i = 1 A i and Λ i = 1 A i respectively, are defined as follows: μ V i ∈ I A i ( x ) = sup { μ A i ( x ) : i ∈ I } , for all x ∈ X and μ Λ i A i ( x ) = inf { μ A i ( x ) : i ∈ I } for all x ∈ X. Throughout this paper X denotes a nonempty fuzzy set. (X,  τ) denotes a fuzzy topological space (fts in short). Throughout x _p denotes a fuzzy point for x ∈ X with the membership value p. The notations fb-open and fθ-open will denote respectively fuzzy b-open and fuzzy θ-open sets in fts X. Let A be a fuzzy subset of X. We denote the fuzzy closure and fuzzy interior of a fuzzy set A by cl (A) and int (A) respectively. A fuzzy point x _p of X is called a fuzzy θ-cluster point of A if cl  (U)  ΛA ≠ 0 _X for every fuzzy open set U of X containing x _p (one may refer to Salleh and Wahab [5]). The set of all fuzzy θ-cluster points of A is called the fuzzy θ-closure of A and is denoted by cl _θ  (A).

A family τ ≤ 1 ^X of fuzzy sets is called a fuzzy topology for X if it satisfies the following three axioms;

0 _X ,  1 _X  ∈ τ.

The pair (X,  τ) is called a fuzzy topological space or fts.

Fuzzy operations: The closer and interior of a fuzzy set A of X are define by Cl ( A ) = inf { K : A ≤ K , K c ∈ τ } . Int ( A ) = sup { O : O ≤ A , O ∈ τ } .

Definition 2.1. [1] A subset S of a space X is called b - open if S ⊂ cl  (int S) ∪ int (cl S).

Definition 2.2. [1] A subset S of a space X is called b - closed if X - S is b-open.

Definition 2.3. [1] A subset S is said to be regular if it is b-open and b-closed.

Definition 2.4. A point x of X is called a θ-cluster point of S if cl (U)∩ S ≠ ∅ for every open set U of X containing x. The set of all θ-cluster point of S is called θ- closure of S and is denoted by cl _θ  (S). A subset S is said to be θ-closed if cl _θ  (S) = S.

The complement of a θ-closed set is said to be θ-open.

3 b-θ open set in fuzzy topological spaces

Definition 3.1. [2] A fuzzy set S in a fts X is called fuzzy b - open (f b - open) set, if and only if S ≤ int (cl S) Vcl (int S).

Definition 3.2. The complement of a fuzzy b-open set is a fuzzy b - closed set, i.e. A is fuzzy b-closed, if X - A is fuzzy b-open.

Definition 3.3. [2] A fuzzy set S is said to be fuzzy b - regular if it is fuzzy b-open and fuzzy b-closed.

Definition 3.4. [5] A fuzzy set is said to be fuzzyθ-closed if S = cl _θ  (S) and fuzzy θ-open if S = int _θ  (S).

The family of all fuzzy b-open (resp b-closed,b-regular) fuzzy set of X is denoted by FBO (X) (resp FBC (X) ,  FBR (X)) and the family of all fuzzyb-open (resp fuzzy b-regular) sets of X containing a fuzzy point x _p  ∈ X is denoted by FBO (X,  x _p ) (resp FBR (X,  x _p )).

A fuzzy point x _p of X is called a fuzzy b-θ-cluster point of fuzzy set A if f bcl (U) ΛA ≠ 0 _X for every U ∈ FBO (X,  x _p ). The set of all fuzzy b-θ-cluster points of A is called fuzzy b-θ-closure of A and it is denoted by f bcl _θ  (A). A subset A is said to be f b-θ-closed set if A = f bcl _θ  (A). The complement of a fuzzy b-θ-closed set is said to be fuzzy b-θ-open.

Theorem 3.1. For any fuzzy subset A of a space X, the following hold: f bcl θ ( A ) = Λ { V : A and V is f b - θ - closed } = Λ { V : A < V and V ∈ BR ( X ) }

Proof. We prove the first equality science the other case can be proved similarly. First suppose that x _p  ∉ f bcl _θ  (A). Then there exists V ∈ FBO (X,  x _p ) such that f bcl (V) ΛA = 0 _X . Since X - f bcl (V) is f b-regular and hence X - f bcl (V) is a fuzzy b-θ-closed set containing A and x _p  ∉ X - f bcl (V). Therefore we have x _p  ∉ Λ {V : A < V and V is f b - θ - closed}.

Conversely, suppose that x _p  ∉ Λ {V : A < Vand V is fb - θ - closed} there exist a f b-θ-closed set V such that A < V and x ∉ V. there exists U ∈ FBO (X) such that x _p  ∈ U < X ∖ V. Therefore we have f bcl (U) Λ A < f bcl (U)  ΛV = 0_X. This shows that x _p  ∉ f bcl _θ  (A).

Theorem 3.2. Let A and B be two subsets of a fts X. Then the following properties hold:

x _p  ∈ f bcl _θ   (A) if and only if VΛA ≠ 0 _X for each V ∈ FBR (X,  x _p ).

Proof. The proof of (a) and (b) are obvious, so omitted.

(c) Generally we have fbcl _θ   (fbcl _θ   (A)) ≥ fbcl _θ  (A) suppose that x _p  ∉ fbcl _θ  (A). There exists V ∈ FBR (X,  x _p ) such that VΛA = 0 _X . Science V ∈ FBR (X), we have VΛfbcl _θ  (A) =0 _X . This shows that x _p  ∉ f bcl _θ   (fbcl _θ   (A)). Therefore we have f bcl _θ   (f bcl _θ   (A)) < f bcl _θ   (A).

(d) Let A _α be f b-θ-closed in X for each α ∈ I, Thus for each α ∈ I, A _α  = f bcl _θ   (A).

Hence we have f bcl _θ   (∩ _α∈IA _α ) < ∩ _α∈Ifbcl _θ  (A _α ) = ∩ _α∈IA _α  < f bcl _θ  (∩ _α∈IA _α ).

Thus, we have f bcl _θ   (∩ _α∈I A _α ) = ∩ _α∈IA _α .

This shows that ∩_α∈I A _α is f b-θ-closed.

Remark 3.1. The union of two fb-θ-closed sets is not necessarily fb-θ-closed, which follows from the following example.

Example 3.1. Let X = {a,  b} and consider the fuzzy subsets on X, A = {a_0.2, b_0.2}, B = {a_0.4, b_0.4}. Let τ = {0 _X ,  1 _X ,  A,  B,  A ∨ B,  A ∧ B}. The sets {A}, {B} are b-θ-closed in (X,  τ) but {A ∨ B} is not b-θ-closed.

Corollary 3.3. Let A and A _α (α ∈ I) be any subsets of a fts X. Then the following properties hold:

A is b-θ-open in X if and only if for each x ∈ A there exists V ∈ BR (X,  x _p ) such that x _p  ∈ V < A.

Theorem 3.3. For a subset A of a fts X, the following properties hold:

If A ∈ FBO (X), then f bcl (A) = f bcl _θ  (A).

Proof. (a) Generally we have f bcl (B) < f bcl _θ  (B) for every subset B of X. Let A ∈ FBO (X) and suppose that x ∉ f bcl (A). Then there exists VΛFBO (X,  x _p ) such that VΛA = 0 _X . Since A ∈ FBO (X), we have f bcl (V) Λ A = 0 _X . This shows that x _p  ∉  f bcl _θ  (A). Hence we obtain f bcl (A) = f bcl _θ  (A).

(b) Let A ∈ FBR (X). Then A ∈ FBO (X) and by (a), A = f bcl (A) = θ - f bcl _θ  (A).

Therefore, A is b-θ-closed set. Since X ∖ A ∈ FBR (X), by the argument above, X ∖ A is b-θ-closed and hence A is b-θ-open. The converse is obvious.

Remark 3.2. It is obvious that b-regular ⇒ b-θ-open ∈ b-open. But the converses are not necessarily true as shown by the following examples.

Example 3.2. Let X = {a,  b} and consider the fuzzy sets on X, are A = {a_0.2,  b_0.2}, B = {a_0.4,  b_0.4}. Let τ = {0 _X ,  1 _X ,  A,  B,  A ∨ B,  A ∧ B}Then the subset set {A ∨ B} is b-θ-open in X but not b-regular.

Example 3.3. Let X = {a,  b} and consider the fuzzy sets on X are A = {a_0.2,  b_0.2}, B = {a_0.4,  b_0.4}, C = {a_0.4,  b_0.9}. Let τ = {0 _X ,  1 _X ,  A,  B,  A ∨ B,  A ∧ B,  A ∧ C}. Then the subset {A ∨ B} is b-open in X but not b-θ-open.