EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 2, 2020, 269-279 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On Generalized β-Open Sets in Ideal Bitopological Space Ibtissam Bukhatwa1,2,∗, Sibel Demiralp1 1 Department of Mathematics, University of Kastamonu, Kastamonu 37150, Turkey 2 Department of Mathematics, University of Benghazi, Benghazi 16063, Libya Abstract. In this article, we introduce and study the concepts of γij-semi-I-open sets and γij- βI-open sets by generalizing (i, j)-semi-I-open sets and (ij)-βI-open sets, respectively, in ideal bitopological spaces with an operation γ : τ → P (X). Further, we describe and study (γ, δ)ij-semi- I-continuous and (γ, δ)ij-βI-continuous functions in ideal bitopological spaces and their related notions. In addition, various examples and counterexamples are given for answers to some questions raised in this study. 2020 Mathematics Subject Classifications: 54A05, 54A10, 54C05, 54E55 Key Words and Phrases: Ideal bitopological space, Intγi(A), Clγi(A),γij-semi-I-open sets, γij-βI-open sets, (γ, δ)ij-βI−continuous functions. 1. Introduction Kelly [11] in 1963, introduced the triple (X, τ1, τ2) as bitopological space, where X is a nonempty set, τ1 and τ2 are topologies on X. Levine [17] in 1963, introduced the notion of semi-open sets in bitopological spaces. Khedr [14] in 1992, defined semi-preopen (β-open)sets in bitopological spaces. K. Kuraowski [15] in 1966, studied and applied the concept of ideals on topological spaces. An ideal I on a topological space (X, τ) is a collection of subsets of X having the heredity property (i) if A ∈ I and B ⊂ A then B ∈ I and (ii) if A ∈ I and B ∈ I then A ∪ B ∈ I. Ekici [5] in 2012, studied the concept of semi-I-open sets in ideal topological spaces. If I is an ideal on X then (X, τ1, τ2, I) is called an ideal bitopological space. Kasahara.S [10] in 1979 described an operation γ on τ as a mapping γ : τ → P (X) such that U ⊆ Uγ , for each U ∈ τ . Khedr [12] in 1984, extended the operation γ to bitopological space as a mapping γ : τ1∪τ2 → P (X) such that for each U ∈ τ1 ∪ τ2, where Uγ denotes the value of γ at U . For example the operations Uγ = U , Uγ = Cli(U), Uγ = Intj(Cli(U)) for U ∈ τj are operations on τ1 ∪ τ2. Caldas [3] in 2013, introduced the notion of β-open sets in ideal bitopological spaces. Csaszar [4] in 1997, defined generalized open sets in generalized topological spaces. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3649 Email addresses: i.bukhatwa@gmail.com (I. Bukhatwa), sdemiralp@kastamonu.edu.tr (S. Demiralp) http://www.ejpam.com 269 c© 2020 EJPAM All rights reserved. I. Bukhatwa, S. Demiralp / Eur. J. Pure Appl. Math, 13 (2) (2020), 269-279 270 2. Preliminaries Throughout the paper, (X, τ1, τ2) always mean bitopological space on with no separa- tion axioms are supposed in this space, also (X, τ1, τ2, I) be an ideal bitopological space. Let A be a subset of X, by Inti(A) [10] and Cli(A) [20] we denote respectively the interior and closure of A with regard to τi for i = 1, 2. A subset A of a bitopological space will be called a γi-open set if for each x ∈ A, there exists an τi-open set U such that x ∈ U and Uγ ⊆ A. Let τγi denotes the set of all γi-open set in X. Obviously, we have τγi ⊆ τi [13]. Complement of all γi-open sets are called γi-closed. Assumed (X, τ1, τ2, I) as an ideal bitopological space and if P (X) is the set of all subsets of X, a set operator (·)∗i : P (X)→ P (X) named the local function of A [22] with regard to τi and τ . The definition of local function is given as: for A ⊂ X, A∗i (τi, I) = {x ∈ X|U ∩ A /∈ I, for all U ∈ τi(x)} where, τi(x) = {U ∈ τi|x ∈ U}. Observe additionally that closure operator for τ∗i (I) accurate than τi is defined by Cl∗i (A) = A∪A∗i . The interior of A in τ∗i (I) is denoted by Int∗i (A) and Int∗γi(A ∗ i ) denotes the interior of A∗i with respect to topology τi, where A∗i = {x ∈ X|U ∩ A /∈ I}, for every U ∈ τi. The interiorγi of A is denoted by Intγi(A) and described to be the union of all γi-open sets of X contained in A. The closureγi of A is denoted by Clγi(A) and defined to be the intersection of all γi-closed sets containing A. Currently, several results and definitions from [2, 3, 7, 13, 17] are recalled to be used in this article. Definition 1. [8] A subset A of a bitopological space (X, τ1, τ2) with operation γ on τ1∪τ2 is named: 1. γij-semi-open set if A ⊆ Clγj (Intγi(A)), where i 6= j and i, j = 1, 2. 2. γij-β-open set if A ⊆ Clγj (Intγi(Clγj (A))), where i 6= j and i, j = 1, 2. Definition 2. [3] A subset A of an ideal bitopological space (X, τ1, τ2, I) is called 1. (i, j)-semi-I−open set if A ⊆ Cl∗j (Inti(A)), where i 6= j and i, j = 1, 2. 2. (i, j)-βI-open set if A ⊆ Clj (Inti(Cl∗j (A))), where i 6= j and i, j = 1, 2. Definition 3. [16] Let (X, τ1, τ2, I) be an ideal bitopological space with an operation γ on τ1∪τ2. The γ-local function of A with regard to γ and I is described as giving, for A ⊂ X, A∗γi(γ, I) = {x ∈ X|U ∩A /∈ I, for every U ∈ τγi(x)} where τγi(x) = {U ∈ τγi |x ∈ U}. In the case of no ambiguity, we will replace A∗γi(γ, I) by A∗γi . Definition 4. [22] Let (X, τ1, τ2, I) be an ideal bitopological space with an operation γ and (Y, σ1, σ2) be a bitopological space with an operation δ. Then a function f : (X, τ1, τ2, I)→ (Y, σ1, σ2) is called pairwise (γ, δ)i-continuous function if f−1(V ) is γi-open in X for all δi-open set V in Y , for i = 1, 2. Definition 5. [3] A function f : (X, τ1, τ2, I) → (Y, σ1, σ2) is called to be (i, j)-semi-I- continuous function (resp.(i, j)-βI−continuous) if f−1(V ) is (i, j)-semi-I-open(resp.(i, j)- βI-open) in X for all γi-open set V in Y , where i 6= j and i, j = 1, 2. Throughout the article, we suppose that i 6= j,and i, j = 1, 2. I. Bukhatwa, S. Demiralp / Eur. J. Pure Appl. Math, 13 (2) (2020), 269-279 271 3. γij-βI−Open Sets This section deals with the concept of γij-βI-open sets and some of their characteri- zations in an ideal bitopological space. Definition 6. A subset A of an ideal bitopological space (X, τ1, τ2, I), with an operation γ on τ1 ∪ τ2, is said to be γij-semi-I-open set if A ⊆ Cl∗γj (Intγi (A)). Example 1. Let X = {a, b, c, d} and (X, τ1, τ2) be a bitopological space with τ1 = {∅, X, {b}, {c, d}, {b, c, d}}, τ2 = {∅, X, {a}, {b}, {a, b}, {a, c, d}} and I = {∅, {a}} and let Uγ = Clj(U) for U ∈ τi. Then we have, γ12-semi-I-open sets are ∅, X, {b}, {c, d}, {b, c, d}. Definition 7. A subset A of an ideal bitopological space (X, τ1, τ2, I) is said to be γij-βI- open set if A ⊆ Clγj (Intγi(Cl∗γj (A))). The set consisting of all γij-βI-open sets in X will be denoted by γij-βIO(X). Definition 8. A subset A of an ideal bitopological space (X, τ1, τ2, I) is called γij-βI-closed set if the complement Ac is a γij-βI-open set. Equivalently, A is called γij-βI-closed set if A ⊇ Intγi(Clγj (Int ∗ γi(A))). The set consisting of all γij-βI-closed sets in X will be denoted by γij-βIC(X). Theorem 1. Let (X, τ1, τ2, I) be an ideal bitopological space, (i) Every γij-semi-I-open set is γij-βI-open. (ii) Every γij-βI-open set is γij-β-open. Proof. (i) Let A be a subset of X. If A is γij-semi-I-open, then A ⊆ Cl∗γj (Intγi(A)) ⊆ Intγi(A) ∪ (Intγi(A))∗γj ⊆ (Intγi(A)) ∪ Clγj (Intγi(A)) ⊆ Clγj (Intγi(A)) ⊆ Clγj (Intγi(A ∪A∗γj )) ⊆ Clγj (Intγi(Cl ∗ γj (A))). Therefore, A is a γij-βI-open set. (ii) Let A be a subset of X. If A is γij-βI-open, then A ⊆ Clγj (Intγi(Cl ∗ γj (A))) ⊆ Clγj (Intγi(A∗γj ∪A)) ⊆ Clγj (Intγi(Clγj (A) ∪A)) ⊆ Clγj (Intγi(Clγj (A))). Therefore, A is a γij-β-open set. But generally the convers of this theorem is not true as giving in the next example. I. Bukhatwa, S. Demiralp / Eur. J. Pure Appl. Math, 13 (2) (2020), 269-279 272 Example 2. From example 1, let A = {c} or A = {b, d}. Calculations show that A is γ12-βI-open, however, it is not γ12-semi-I-open. Conclusion 1. Let (X, τ1, τ2, I) be an ideal bitopological space. Then every γij-semi-I- open set is γij-β-open. Theorem 2. Let (X, τ1, τ2, I) be an ideal bitopological space. Then (i) The union of any γij-βI-open sets is γij-βI-open set. (ii) The intersection of any γij-βI-closed sets is γij-βI-closed set. Proof. (i) Let Aα ∈ γij-βIO(X) for each α ∈ Λ, where Λ is an index set. Then Aα ⊆ Clγj (Intγi(Cl∗γj (Aα))). Therefore, ∪α∈ΛAα ⊆ ∪α∈Λ{Clγj (Intγi(Cl∗γj (Aα)))} ⊆ {Clγj (Intγi(∪α∈ΛCl ∗ γj (Aα)))} ⊆ {Clγj (Intγi(Cl∗γj (∪α∈ΛAα)))}. Then ∪α∈ΛAα is γij-βI-open. (ii) The proof follows by using (i) and taking complement. The intersection of any two γij-βI-open sets may not be an γij-βI-open set as showing in the next example. Example 3. Let X = {a, b, c, d}, τ1 = {∅, X, {a}, {d}, {a, d}, {a, c, d}}, τ2 = {∅, X} and I = {∅, {b}, {c}, {b, c}}. Let define an operation γ : τ1 ∪ τ2 → P (x) such that Uγ = U for all U ∈ τi. Then we have {a, c} and {c, d} are γ12-βI-open sets but {c} is not γ12-βI-open. Definition 9. Let (X, τ1, τ2, I) be an ideal bitopological space with an operation γ, A ⊂ X and x be a point of X. Then (i) x is called an βI-interiorγij point of A if there exists any U ∈ γij-βIO(X) such that x ∈ U ⊂ A. (ii) The set of all βI-interiorγij points of A is called γij-βI-interior of A and is repre- sented by βI-Intγij (A). Theorem 3. Let A and B be subsets of (X, τ1, τ2, I). Then the following properties hold: 1) βI-Intγij (A) = ∪{U : U ⊂ A and U ∈ γij-βIO(X)}. I. Bukhatwa, S. Demiralp / Eur. J. Pure Appl. Math, 13 (2) (2020), 269-279 273 2) βI-Intγij (A) is the largest γij-βI-open subset of X contained in A. 3) A is γij-βI-open if and only if A = βI-Intγij (A). The proof will be obtained directly from the definition and thus the proof is omitted. Definition 10. Let (X, τ1, τ2, I) be an ideal bitopological space with an operation γ, A ⊂ X and x be a point of X. Then, 1. x is called an γij-βI-cluster point of A if U ∩ A 6= ∅ for every U ∈ γij-βIO(X) such that x ∈ U . 2. The set of all γij-βI-cluster points of A is called γij-βI-cluster of A and is repre- sented by βI-Clγij (A). Theorem 4. Let A and B be subsets of (X, τ1, τ2, I). Then the following properties hold: 1) βI-Clγij (A) = ∩{V : A ⊂ V and V ∈ γij-βIC(A). 2) βI-Clγij (A) is the smallest γij-βI-closed subset of X containing A. 3) A is γij-βI-closed if and only if A = βI-Clγij (A) The proof will be obtained directly from the definition and thus the proof is omitted. Theorem 5. Let (X, τ1, τ2, I) be an ideal bitopological space with an operation γ and A ⊂ X. Then, (i) If I = {∅}, then A is γij-βI-open if and only if A is γij-β-open. (ii) If I = P (X), then A is γij-βI-open if and only if A is γij-semi-open. Proof. (i) We have just to show that if I = {∅} and A is γij-β-open, then A is γij-βI-open. If I = {∅}, then A∗γj = Clγj (A) for all subset A of X. Assumed A to be γij-β-open set, then A ⊆ Clγj (Intγi(Clγj (A))) ⊆ Clγj (Intγi(A∗γj )) ⊆ Clγj (Intγi(A ∗ γj ∪A)) ⊆ Clγj (Intγi(Cl∗γj (A))). Therefore, A is γij-βI-open. (ii) Let I = P (X), then A∗γj = {∅} for any subset A of X. Let A be γij-semi-open. Then A ⊆ Clγj (Intγi(A)) = Clγj (Intγi(A∪A∗γj )) = Clγj (Intγi(Cl ∗ γj (A))). Therefore, A is γij-βI -open. I. Bukhatwa, S. Demiralp / Eur. J. Pure Appl. Math, 13 (2) (2020), 269-279 274 Theorem 6. Let (X, τ1, τ2, I) be an ideal bitopological space with an operation γ and A ⊂ X. Then A is γij-βI-open if and only if Clγj (A) = Clγj (Intγi(Cl ∗ γj (A))). Proof. Let A be an γij-βI-open subset of X. Then A ⊆ Clγj (Intγi(Cl ∗ γj (A))). Hence Clγj (A) ⊆ Clγj (Intγi(Cl∗γj (A))). Since A∗γj ∪A ⊆ Clγj (A), then we have, Clγj (A) ⊆ Clγj (Intγi(Cl∗γj (A))) ⊆ Clγj (Intγi(Clγj (A))) ⊆ Clγj (A). Therefore, Clγj (A) = Clγj (Intγi(Cl ∗ γj (A))). The convers is obvious. 4. (γ, δ)ij-βI-Continuous Functions In this section the concept of (γ, δ)ij-βI-continuous function in ideal bitopological spaces are introduced along with some characterizations via related notions. Throughout this section, let (X, τ1, τ2, I) be an ideal bitopological space with an op- eration γ, and (Y, σ1, σ2) be a bitopological space with an operation δ. Definition 11. A function f : (X, τ1, τ2, I)→ (Y, σ1, σ2) is called (γ, δ)ij-semi-I-continuous function (resp. (γ, δ)ij-βI continuous) if f−1(V ) is γij-semi-I-open (resp. γij-βI open) in X for all δi-open set V in Y . Generally every (γ, δ)ij-semi-I-continuous function is (γ, δ)ij-βI-continuous, but the convers is not true as giving in next example. Example 4. Let X = {a, b, c, d} be a set and (X, τ1, τ2, I) be an ideal bitopological space with τ1 = {∅, X, {b}, {c, d}, {b, c, d}}, τ2 = {∅, X, {a}, {a, b}, {a, c, d}}, I = {∅, {b}}, Uγ = Clj (U) for U ∈ τi. Let Y = {p, q, r, s} be a set and (Y, σ1, σ2) be a bitopological space with σ1 = {∅, Y, {q}, {r, s}, {q, r, s}}, σ2 = {∅, Y, {p}, {p, q}, {p, r, s}}, V δ = V for V ∈ σi. Let define f : (X, τ1, τ2, I) → (Y, σ1, σ2) such that f(a) = p, f(b) = q, f(c) = r and f(d) = s. Then f is (γ, δ)12-βI-continuous but not (γ, δ)12-semi-I-continuous because {p} is δi-open set and f−1({p}) = {a} which is γ12-βI-open in X but not γ12-semi-I-open in X. Theorem 7. For any function f : (X, τ1, τ2, I) → (Y, σ1, σ2), the next properties are equivalent, 1) f is (γ, δ)ij-βI-continuous 2) For all x ∈ X and every δi-open set V in Y containing f(x), there exists a γij-βI- open set U of X containing x such that f(U) ⊂ V . I. Bukhatwa, S. Demiralp / Eur. J. Pure Appl. Math, 13 (2) (2020), 269-279 275 Proof. (1⇒ 2) Let V is δi-open in Y such that f(x) ∈ V . Since f is (γ, δ)ij-βI-continuous, f−1(V ) is γij-βI-open set in X. Let U = f−1(V ). Then f(x) ∈ f(U) ⊂ V. (2⇒ 1) Let V be δi-open set in Y and x ∈ f−1(V ). Then V is δi-open set in Y and f(x) ∈ V . From the hypothesis, there exists an γij-βI-open set U in X containing x such that f(U) ⊂ V . Then x ∈ U ∈ f−1(V ), i.e. f−1(V ) is γij-βI-open set in X. Therefore, f is (γ, δ)ij-βI-continuous. Theorem 8. Let f : (X, τ1, τ2, I)→ (Y, σ1, σ2) be a (γ, δ)ij-βI-continuous function. Then the next properties are equivalent: 1) The inverse image of every δi-closed set in Y is γij-βI-closed set in X, 2) f(βI-Clγij (U)) ⊂ Clδj (f(U)), for all subset U of X, 3) βI-Clγij (f −1(V )) ⊂ f−1(Clδj (V )), for each subset V of Y . Proof. (1⇒ 2) Let U ⊂ X. Since Clδj (f(U)) is an δi-closed set in Y , from the hypothesis, we have f−1(Clδj (f(U))) is γij-βI-closed set in X. Also U ⊂ f−1(Clδj (f(U))) and βI-Clδi(U) is the smallest γij-βI-closed set containing U . Therefore, βI − Clδi(U) ⊂ f−1(Clδi(f(U))). This implies that f(βI-Clγij (U)) ⊂ Clδi(f(U)). (2⇒ 3) Let V ⊂ Y . Then f−1(V ) ⊂ X. From the hypothesis, f(βI − Clγij (f−1(V ))) ⊂ Clδj (f(f−1(V ))) ⊂ Clδj (V ). Hence βI-Clγij (f −1(V )) ⊂ f−1(Clδj (V )). (3⇒ 1) Let V be a δi-closed set in Y . From the hypothesis, βI − Clδi(f −1(V )) ⊂ f−1(Clδj (V )) = f−1(V ). Therefore, f−1(V ) = βI-Clγij (f −1(V )) and so f−1(V ) is γij-βI-closed set in X. Theorem 9. The function f : (X, τ1, τ2, I) → (Y, σ1, σ2) is (γ, δ)ij-βI-continuous func- tion if and only if f−1(Intδi(V )) ⊂ βI − Intγijf−1(V ) for all δi-open set of Y. I. Bukhatwa, S. Demiralp / Eur. J. Pure Appl. Math, 13 (2) (2020), 269-279 276 Proof. Let f is a (γ, δ)ij-βI-continuous function and V be an δi-open set in Y . Then f−1(Intδi(V )) is a γij-βI-open set in X. Therefore, f−1(Intδi(V )) ⊂ βI − Intγijf−1(Intδi(V )) ⊂ βI − Intγijf−1((V )). If f−1(Intδi(V )) ⊂ βI-Intγijf −1((V )) and V be a δi-open set of Y , then f−1(V ) = f−1(Intδi(V )) ⊂ βI − Intγijf−1((V )). Therefore, f−1(V ) is γij-βI-open set in X and so f is a (γ, δ)ij-βI-continuous function. Note that, the composition of two (γ, δ)ij-βI-continuous functions need not to be (γ, δ)ij-βI-continuous, in general. Example 5. Let (X, τ1, τ2, I) and (X,σ1, σ2, ζ) be two ideal bitopological spaces such that X = {a, b, c}, τ1 = {∅, X, {a, b}}, τ2 = {∅, X, {a}, {a, b}}, I = {∅, {b}}, Uγ = { Clj(U) for U ∈ τi, b /∈ U U , b ∈ U and σ1 = {∅, X, {b}, {b, c}}, σ2 = {∅, X, {b, c}}, ζ = {∅, {a}}, V δ = V for V ∈ σi. Let define f : (X, τ1, τ2, I) → (X,σ1, σ2) such that f(a) = b, f(b) = a and f(c) = c and let (X, ς1, ς2) be a bitopological space such that ς1 = {∅, X, {a}, {c}{a, c}}, ς2 = {∅, X, {b, c}}, W ξ = { Clj(W ) for W ∈ τi, c /∈ W W , c ∈ W and define g : (X, τ1, τ2, I) → (X, ς1, ς2) such that g(a) = b, g(b) = c and g(c) = a. Then f is (γ, δ)12-βI-continuous function and g is (δ, ξ)12-βI-continuous function but the composition g ◦ f is not (γ, ξ)12-βI−continuous function because {a} is ξ1-open set and (g ◦ f)−1({a}) = {c} /∈ γ12-βIO(X). Definition 12. A function (X, τ1, τ2, I) → (Y, σ1, σ2) is said to be pairwise (γ, δ)i-βI- continuous function if f−1(V ) is γi-βI-open in X for every δi-open set V in Y . Note that the notion of pairwise (γ, δ)i-βI-continuous and (γ, δ)ij-βI-continuous are independent. Example 6. Let X = {a, b, c}, τ1 = {∅, X, {b}, {b, c}}, τ2 = {∅, X, {b, c}}, I = {∅, {a}} with operation Uγ = U for U ∈ τi and let (X,σ1, σ2) be a bitopological space such that σ1 = {∅, X, {a}, {c}, {a, c}}, σ2 = {∅, X, {b, c}} with operation V δ = { Clj(V ) for V ∈ σj , c /∈ V V , c ∈ V and define f : (X, τ1, τ2, I) → (X,σ1, σ2) such that f(a) = b, f(b) = c and f(c) = a Then f is (γ, δ)12-βI-continuous function but it is not (γ, δ)1-βI-continuous function since {a} ∈ δ1-open and f−1({a}) = {c} which is not (γ, δ)1-βI-open. REFERENCES 277 Theorem 10. Let f : (X, τ1, τ2, I)→ (Y, σ1, σ2) and g : (X, τ1, τ2, I)→ (Z, `1, `2). Then g ◦ f is (γ, ξ)ij-βI-continuous if f is (γ, δ)ij-βI-continuous and g is pairwise (δ, ξ)i-βI- continuous. Proof. Let W ∈ ξi-open set in Z. Since g is pairwise (δ, ξ)i-continuous, then g−1(W ) ∈ δi-open set in Y . On the other hand, since f is (γ, δ)ij-βI-continuous, f−1(g−1(W )) ∈ γij- βIO(X). Therefore, we obtain that g ◦ f is (γ, ξ)ij-βI-continuous. 5. Conclusion In this study, we defined the notion of γij-semi-I-open sets and γij-βI-open sets by generalizing (i, j)-semi-I-open sets and (ij)-βI-open sets in ideal bitopological spaces with an operation γ : τ → P (X). We show that every γij-semi-I-open set is a γij-βI- open but the converse is not always true. Then we described the notions γij-βI-interior and γij-βI-cluster of a set A. Finally we characterized (γ, δ)ij-semi-I-continuous and (γ, δ)ij-βI-continuous functions and showed that any (γ, δ)ij-semi-I-continuous function is a (γ, δ)ij-βI-continuous but the converse is not always true. Also it is shown that the composition of two (γ, δ)ij-βI-continuous functions need not to be (γ, δ)ij-βI-continuous. Consequently the following diagrams are true: γij-semi-I-open −→ γij-βI-open −→ γij-β-open γij-βI-open←→ γij-β-open (I = {∅}) γij-βI-open←→ γij-semi-open (I = P (X)) (γ, δ)ij-semi-I-continuous −→ (γ, δ)ij-βI-continuous These notations, defined in this study, can be extended to other practicable research fields of topology such as fuzzy topology, soft topology, intuitionistic topology and so on. Also generalized seperation axioms can be introduced by the concept of generalized β-open set. References [1] B. Ahmad and S. Hussain. γ-Semi-Open Sets in Topological Spaces II. Southeast Asian Bull. Math 34, no. 6 ,997-1008, 2010. [2] B. Bhattacharya and A. Paul. A New Approach of γ-open Sets in Bitopological Spaces, Gen. Math. Notes 20(2), 95-110, 2014. [3] M. Caldas, S. Jafari and N. Rajesh. Some Fundamental Properties of β-open Sets in Ideal Bitopological Spaces. European Journal of Pure and Applied Mathematics, 6(2), 247-255, 2013. [4] A.Csaszar. Generalized open sets. 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