A Note on (i,j)-πgβ Closed Sets in Intuitionistic Fuzzy Bitopological Spaces

In this paper we introduce the concept of (i,j)–πgβ-closed set in intuitionistic fuzzy bitopological spaces as a generalization of πgβ-closed set in fuzzy bitopological space and study their related notions in bitopological spaces. Next, we introduce (i,j)–πgβopen sets in intuitionistic fuzzy bitopological spaces, and investigate some of their basic properties. Using these concepts, the characterizations for the intuitionistic fuzzy pairwise (i,j)–πgβ continuous mappings are obtained. The relationships between intuitionistic fuzzy pairwise (i,j)–πgβ continuous mappings are discussed. Finally, we prove the irresoluteness in (i,j)–πgβ intuitionistic fuzzy bitopological spaces.


Introduction
The notion of β-open set was introduced by Abd El-Monsef et al. [1] and Andrijevic [2]. Later on, as a generalization of the above mentioned set, πgβ sets have been introduced by Caldas and Jafari [4]. The concept of bitopological spaces (X,τ i ,τ j ) was introduced by Kelly J. C in 1963 [8] where X is a nonempty set and the bitopological spaces are equipped with two arbitrary topologies τ i and τ j , Where τ i and τ j are two topologies on X. After that several authors turned their attention towards generalizations of various concepts of topology for bitopological spaces. In 2005, the concept of (i,j)-β-open sets was defined and investigated by Raja Rajeswari and Lellis Thivagar [9]. As a generalization of fuzzy sets, the concept of intuitionistic fuzzy sets was introduced by Atanassov [3]. Recently, Coker [5] introduced intuitionistic fuzzy topological spaces using intuitionistic fuzzy sets. On the other hand, Kandil [7] introduced the concept of fuzzy bitopological spaces as a natural generalization of Chang's fuzzy topological spaces.
In 2012, the notion of bitopological space was introduced in intuitionistic fuzzy topology by -Jin Tae Kim, Seok Jong Lee [6]. In this paper, the concepts of πgβ-closed set have been extended to the bitopological spaces in intuitionistic fuzzy topology and we introduce a new form of closed set called Intuitionistic fuzzy (IF) (i,j)-πgβ-closed set., The notion of IF (i,j)-πgβ -continuous function and irresolute function is introduced and studied.

Preliminaries
The interior and the closure of a subset A of an intuitionistic fuzzy bi topological space (IFBTS) (X,τ) are denoted by Int(A) and Cl(A), respectively.
In the following sections by X, Y and Z, we mean an intuitionistic fuzzy bi topological space (X,τ i, τ j ), (Y, σi,σ j ) and (Z,η i ,η j ), respectively.
For, a subset A of a bi topological space (X,τi, τj), we denote the closure of A and the interior of A with respect to τ i by τ i-Cl(A) and τ i -Int(A), respectively. Definition 2.1. Let A be an intuitionistic fuzzy set in an IFBTS (X,τ i, τ j ), Then A is said to be an (i) IF (i, j)-semi open [6] if there exists an IF,τ i -open set U in X such that U ⊆ A⊆ j -Cl(U) and IF (i, j)-semi closed [6] if there exists an IF,τ i -closed set U in X such that j-Int(U) ⊆ A ⊆ U.
Then A⊆X\U and X\U is IF(i, j) πgβ-closed.
Then there exists U∈(j,i) πgβO(X, x) such that U∩A=∅. This is a contradiction to U∩A ≠ ∅, Hence x∈IF(j,i) πgβ-
(ii) Similar to (i)

Definition 3.3
A space X is said to be IF(i,j)-πgβT 1 if for any two distinct points x, y of X, there exists IF (i, j)-πgβ open sets U, V such that x∈U but y∉U and y ∈V but x∉V.
Theorem 3.2 An intuitionistic fuzzy bitopological space X is IF (i,j) πgβT 1 if and only if {x} is IF(i,j)-πgβ closed in X for every x∈X.
Proof. If {x} is IF (i, j) πgβ closed in X for every x∈X, for x≠y, X\{x}, X\{y} are IF(i, j)-πgβ open sets such that y∈X\{x} and x∈X\{y}. Therefore, X is IF(i, j) πgβT 1 .
Conversely, if X is IF (i,j)-πgβT 1 , if and if y∈X\{x} then x≠y. Therefore, there exist IF(i,j) πgβ open sets Ui, Vj ∈X such that x∈U i but y ≠Ui and y∈Vj but x∉Vj.
Let G be the union of all such Vj. Then G is an IF (i, j) πgβ-open set and G⊂X\{x}⊂X.
Therefore, X\{x} is an IF (i, j) πgβ open set in X. Theorem 3.3: If A is IF(i,j) πgβ closed and A⊆B⊆(j,i) πgβ-Cl(A),then B is also IF(i,j) πgβ closed set. Hence A∪(X-IF(i, j)πgβ-Cl(A)) is a IF(i, j)-πgβclosed set.
Theorem 3.6. Let A be an IF(i,j)-πgβ-closed in intuitionistic fuzzy bitoplogical space X. Then IF(j,i) β-Cl(A)\A does not contain any nonempty IF(i,j) π-closed set.
Proof. Let U be a nonempty IF (i,j) π-closed subset of Proof. Necessity: Let A be an IF(i, j) πgβ-closed, By Proof. Here, B⊆A and A is both a IF(i,j) πgβ -closed and IF(i,j) open set, then IF(j,i) βCl(A)⊆A and thus IF (j,i) β- Conversely if B is IF (i,j)πgβ-closed in X and U is an Intuitionistic fuzzy bitopological π-open subset of A such that B⊂U, then U=V∩A for some open subset V of X. As B⊆V and B is IF(i,j) πgβ-closed in X,

πgβCl-{y}
Hence there exists an IF(i, j) πgβ -open set containing z and, x but not y. Therefore, y ∉IF (i, j) πgβker{x} and IF (i, j) πgβker{x} ≠ IF(i, j) πgβker{y} Definition 3.6 A space X is said to be IF (i, j)-πgβR 0 if every IF(i, j)-πgβ-open set contains the IF(i, j)-πgβ-closure of each of its singletons.
Lemma 3.4 A space X is IF(i, j)-πgβ R 0 if and only if for any x and y in X and IF(i,j)-πgβcl{x} ≠IF (i, j)-πgβcl{y} implies and IF(i, j)-πgβcl{x} ∩ IF(i, j)-πgβcl{y}=ϕ.
Proof. Let

Conclusion
In this paper, we introduce the concept of πgβ closed set in intuitionistic fuzzy bitopological spaces and study some of their properties. We also introduce the concept of πgβ continuous functions in bitopological spaces and some of their properties have been established. We hope that the findings in this paper are just the beginning of a new structure and not only will form the theoretical basis for further applications of topology on Intuitionistic fuzzy bi topological sets but also will lead to the development of information system and various fields in engineering.