() CUBO A Mathematical Journal Vol.17, No¯ 03, (43–51). October 2015 (i, j)-ω-semiopen sets and (i, j)-ω-semicontinuity in bitopological spaces Carlos Carpintero & Ennis Rosas Department of Mathematics, Universidad De Oriente, Nucleo De Sucre Cumana, Venezuela. Facultad de Ciencias Basicas, Universidad del Atlantico, Barranquilla, Colombia. carpintero.carlos@gmail.com, ennisrafael@gmail.com Sabir Hussain Department of Mathematics, College of Science, Qassim University, P.O.BOX 6644, Buraydah 51482, Saudi Arabia. sabiriub@yahoo.com, sh.hussain@qu.edu.sa ABSTRACT The aim of this paper is to introduce and characterize the notions of (i, j)-ω-semiopen sets as a generalization of (i, j)-semiopen sets in bitopological spaces. We also define and discuss the properties of (i, j)-ω-semicontinuous functions. RESUMEN El objetivo de este art́ıculo es introducir y caracterizar las nociones de conjuntos (i, j)- ω-semiabiertos como una generalización de conjuntos (i, j)-semiabiertos en espacios bitopológicos. También definimos y discutimos las propiedades de funciones (i, j)-ω- semicontinuas. Keywords and Phrases: Bitopological spaces, (i, j)-ω-semiopen sets, (i, j)-ω-semiclosed sets. 2010 AMS Mathematics Subject Classification: 54A05,54C05,54C08. 44 Carlos Carpintero, Sabir Hussain & Ennis Rosas CUBO 17, 3 (2015) 1 Introduction and Preliminaries The concept of a bitopological space was introduced by Kelly [3]. On the other hand, S. Bose [1], introduced the concept of (i, j)-semiopen sets in bitopological spaces. Recently, as generalization of closed sets, the notion of ω-closed sets was introduced and studied by Hdeib [2]. A point x ∈ X is called a condensation point of A, if for each U ∈ τ with x ∈ U, the set U ∩ A is uncountable. A is said to be ω-closed [2], if it contains all of its condensation points. The complement of a ω-closed set is said to be ω-open. It is well known that a subset W of a space (X, τ) is ω-open if and only if for each x ∈ W, there exists U ∈ τ such that x ∈ U and U\W is countable. In this paper, we introduce the concept of (i, j)-ω-semiopen sets as a generalization of (i, j)-semiopen sets in bitopological spaces. We also define and discuss the properties of (i, j)-ω-semicontinuous functions. For a subset A of X, the closure of A and the interior of A are denoted by Cl(A) and Int(A), respectively. A subset A of a bitopological space (X, τ1, τ2) is said to be (i, j)-semi open, if A ⊆ τi-cl(τj-Int(A)), where i 6= j, i, j = 1, 2. The complement of a (i, j)-semiopen set is said to be a (i, j)-semiclosed. The (i, j)-semiclosure of A, denoted by (i, j)-scl(A) is defined by the intersection of all (i, j)-semiclosed sets containing A. The (i, j)-semi interior of A, denoted by (i, j)-sInt(A) is defined by the union of all (i, j)-semiopen sets contained in A. The family of all (i, j)-semiopen (respectively (i, j)-semiclosed) subsets of a space (X, τ1, τ2) is denoted by (i, j) − SO(X), (respectively (i, j) − SC(X)). A function f : (X, τ1, τ2) 7→ (Y, σ1, σ2) is said to be (i, j)-semi continuous, if the inverse image of every σi-open set in (Y, σ1, σ2) is (i, j)-semi open in (X, τ1, τ2), where i 6= j, i, j = 1, 2. A σi-open set U in (Y, σ1, σ2) means U ∈ σi. 2 (i, j)-ω-semiopen sets A set X equipped with two topologies is called a bitopological space. Throughout this paper, spaces (X, τ1, τ2) (or simply X) always means a bitopological spaces on which no separation axioms are assumed unless explicitly stated. Definition 2.1. Let X be a bitopological space and A ⊆ X. Then A is said to be (i, j)-ω-semiopen, if for each x ∈ A there exists a (i, j)-semiopen Ux containing x such that Ux − A is a countable set. The complement of a (i, j)-ω-semiopen set is a (i, j)-ω-semiclosed set. The family of all (i, j)-ω-semiopen (respectively (i, j)-ω-semiclosed) subsets of a space (X, τ1, τ2) is denoted by (i, j) − ω − SO(X), (respectively (i, j) − ω − SC(X)). Also the family of all (i, j) − ω- semiopen sets of (X, τ1, τ2) containing x is denoted by (i, j) − ω − SO(X, x). Note that every (i, j)-semiopen set is a (i, j)-ω-semiopen. The following example shows that the converse is not true in general. Example 2.2. Let X = {a, b, c}, τ1 = {∅, {a, b}, X}, τ2 = {∅, {b, c}, X}. Then {a, c} is a (i, j)-ω- semiopen but not (i, j)-semiopen. CUBO 17, 3 (2015) (i, j)-ω-semiopen sets and (i, j)-ω-semicontinuity . . . 45 Theorem 2.3. Let X be a bitopological space and A ⊆ X. Then A is said to be (i, j)-ω-semiopen if and only if for every x ∈ A, there exists a (i, j)-semiopen set Ux containing x and a countable subset C such that Ux − C ⊆ A. Proof. Let A be a (i, j)-ω-semiopen set and x ∈ A, then there exists a (i, j)-semiopen subset Ux containing x such that Ux − A is countable. Let C = Ux − A = Ux ∩ (X − A). Then Ux − C ⊆ A. Conversely, let x ∈ A. Then there exists a (i, j)-ω-semiopen subset Ux containing x and a countable subset C such that Ux − C ⊆ A. Thus Ux − A ⊆ C and Ux − A is countable. Theorem 2.4. Let X be a bitopological space and C ⊆ X. If C is a (i, j)-ω-semiclosed set, then C ⊆ K ∪ B, for some (i, j)-ω-semiclosed subset K and a countable subset B. Proof. If C is a (i, j)-ω-semiclosed set, then X−C is a (i, j)-ω-semiopen set and hence by Theorem 2.3, for every x ∈ X − C, there exists a (i, j)-semiopen set U containing x and a countable set B such that U − B ⊆ X − C. Thus C ⊆ X − (U − B) = X − (U ∩ (X − B)) = (X − U) ∪ B, let K = X − U. Then K is a (i, j)-semiclosed set such that C ⊆ K ∪ B. Theorem 2.5. The union of any family of (i, j) − ω-semiopen sets is (i, j)-ω-semiopen set. Proof. If {Aα : α ∈ I} is a collection of (i, j)-ω-semiopen subsets of X, then for every x ∈ ⋃ α∈I Aα, x ∈ Aα, for some α ∈ I. Hence, there exists a (i, j)-ω-semiopen subset U containing x, such that U − Aα is countable. Now as U − ( ⋃ α∈I Aα) ⊆ U − Aα, and thus U − ( ⋃ α∈I Aα) is countable. Therefore ⋃ α∈I Aα is a (i, j)-ω-semiopen set. Definition 2.6. The union of all (i, j)-ω-semiopen sets contained in A ⊆ X is called the (i, j)-ω- semi-interior of A and is denoted by (i, j) − ω-SInt(A). The intersection of all (i, j)-ω-semiclosed sets of X containing A is called the (i, j)-ω-semiclosure of A and is denoted by (i, j)-ω − SCl(A). Remark 2.7. The (i, j)-ω-SCl(A) is a (i, j)-ω-semiclosed set and the (i, j)-ω-SInt(A) is a (i, j)- ω-semiopen set. Theorem 2.8. Let X be a bitopological space and A, B ⊆ X. Then the following properties hold: (1) (i, j)-ω-SInt((i, j)-ω-SInt(A)) = (i, j)-ω-SInt(A). (2) If A ⊂ B, then (i, j)-ω-SInt(A) ⊂ (i, j)-ω-SInt(B). (3) (i, j)-ω-SInt(A ∩ B) ⊂ (i, j)-ω-SInt(A) ∩ (i, j) − ω − SInt(B). (4) (i, j)-ω-SInt(A) ∪ (i, j)-ω-SInt(B) ⊂ (i, j)-ω-SInt(A ∪ B). (5) (i, j)-ω-SInt(A) is the largest (i, j)-ω-semiopen subset of X contained in A. (6) A is (i, j)-ω-semiopen if and only if A = (i, j)-ω-SInt(A). (7) (i, j)-ω-SCl((i, j)-ω-SCl(A)) = (i, j)-ω-SCl(A). 46 Carlos Carpintero, Sabir Hussain & Ennis Rosas CUBO 17, 3 (2015) (8) If A ⊂ B, then (i, j)-ω-SCl(A) ⊂ (i, j)-ω-SCl(B). (9) (i, j)-ω-SCl(A) ∪ (i, j)-ω-SCl(B) ⊂ (i, j)-ω-SCl(A ∪ B). (10) (i, j)-ω-SCl(A ∩ B) ⊂ (i, j)-ω-SCl(A) ∩ (i, j)-ω-SCl(B). Proof. (1), (2), (6), (7) and (8) follow directly from the definition of (i, j)-ω-semiopen and (i, j)- ω-semiclosed sets. (3), (4) and (5) follow from (2). (9) and (10) follow by applying (8). Example 2.9. Let X be the real line, τ1 = {∅, Re, Q c} and τ2 = {∅, Re, Q, Q c}. Take A = (0, 1), B = (1, 2), then (i, j)-ω-SCl(A ∩ B) ⊂ (i, j)-ω-SCl(A) ∩ (i, j)-ω-SCl(B). Example 2.10. Let X be the real line, τ1 = {∅, Re, Q} and τ2 = {∅, Re, Q}. The collection of (i, j) − SO(X) is {∅, Re, Q}. take A = Q, B = {π}. Then (i, j)-ω-SCl(A) = Q, (i, j)-ω-SCl(B) = {π} and (i, j)-ω-SCl(A) ∪ (i, j)-ω-SCl(B) ⊂ (i, j)-ω-SCl(A ∪ B). Theorem 2.11. Let X be a bitopological space. Suppose A ⊆ X and x ∈ X. Then x ∈ (i, j)-ω- SCl(A) if and only if U ∩ A 6= ∅ for every U ∈ (i, j)-ω-SO(X, x). Proof. Suppose that x ∈ (i, j)-ω-SCl(A) and we show that U ∩ A 6= ∅, for all U ∈ (i, j)-ω- SO(X, x). Suppose on the contrary that there exists U ∈ (i, j)-ω-SO(X, x) such that U ∩ A = ∅, then A ⊆ X − U and X − U is a (i, j)-ω-semiclosed set. This follows that (i, j)-ω-SCl(A) ⊆ (i, j)- ω-SCl(X − U) = X − U. Since x ∈ (i, j)-ω-SCl(A), we have x ∈ X − U and hence x /∈ U. Which contradicts the fact that x ∈ U. Therefore, U ∩ A 6= ∅. Conversely, suppose that U ∩ A 6= ∅ for every U ∈ (i, j)-ω-SO(X, x). We shall prove that x ∈ (i, j)-ω-SCl(A). Suppose on the contrary that x /∈ (i, j)-ω-SCl(A). Let U = X − (i, j)-ω-SCl(A), then U ∈ (i, j)-ω-SO(X, x) and U ∩ A = (X − ((i, j)-ω-SCl(A))) ∩ A ⊆ (X − A) ∩ A = ∅. This is a contradiction to the fact that U ∩ A 6= ∅. Hence x ∈ (i, j)-ω-SCl(A). Theorem 2.12. Let X be a bitopological space and A ⊂ X. Then the following properties hold: (1) (i, j)-ω-SCl(X\A) = X\(i, j)-ω-SInt(A); (2) (i, j)-ω-SInt(X\A) = X\(i, j)-ω-SCl(A). Proof. (1). Let x ∈ X\(i, j)-ω-SCl(A). Then there exists V ∈ (i, j)-ω-SO(X, x) such that V ∩ A = ∅ and hence we obtain x ∈ (i, j)-ω-SInt(X\A). This shows that X\(i, j)-ω-SCl(A) ⊂ (i, j)-ω- SInt(X\A). Now consider x ∈ (i, j)-ω-SInt(X\A). Since (i, j)-ω-SInt(X\A) ∩ A = ∅, we obtain x /∈ (i, j)-ω-SCl(A). Therefore, we have, (i, j)-ω-SCl(X\A) = X\(i, j)-ω-SInt(A). (2). Let x ∈ X\(i, j)-ω-SInt(X−A). Since (i, j)-ω-SInt(X\A)∩A = ∅, we have x /∈ (i, j)-ω-SCl(A) implies x ∈ X\(i, j)-ω-SCl(A). Now consider x ∈ X\(i, j)-ω-SCl(A), then there exists V ∈ (i, j)-ω- SO(X, x) such that V ∩ A = ∅, hence we obtain that (i, j)-ω-SInt(X\A) = X\(i, j)-ω-SCl(A). Definition 2.13. Let X be a bitopological space and B ⊆ X. Then B is a (i, j)-ω-semineighbourhood of a point x ∈ X if there exists a (i, j)-ω-semiopen set W such that x ∈ W ⊂ B. CUBO 17, 3 (2015) (i, j)-ω-semiopen sets and (i, j)-ω-semicontinuity . . . 47 Theorem 2.14. Let X be a bitopological space and B ⊆ X. B is a (i, j)-ω-semiopen set if and only if it is a (i, j)-ω-semineighbourhood of each of its points. Proof. Let B be a (i, j)-ω-semiopen set of X. Then by definition B is a (i, j)-ω-semineighbourhood of each of its points. Conversely, suppose that B is a (i, j)-ω-semineighbourhood of each of its points. Then for each x ∈ B, there exists Sx ∈ (i, j)-ω-SO(X, x) such that Sx ⊂ B. Then B = ⋃ {Sx : x ∈ B}. Since each Sx is a (i, j)-ω-semiopen and arbitrary union of (i, j)-ω-semiopen sets is (i, j)-ω-semiopen, B is a (i, j)-ω-semiopen in X. Theorem 2.15. If each nonempty (i, j)-ω-semiopen set of a bitopological space X is uncountable, then (i, j)-SCl(A) = (i, j)-ω-SCl(A), for each subset A ∈ τ1 ∩ τ2. Proof. Clearly (i, j)-ω-SCl(A) ⊆ (i, j)-SCl(A). On the other hand, let x ∈ (i, j)-SCl(A) and B be a (i, j)-ω-semiopen set containing x. Using Theorem 2.3, there exists a (i, j)-semiopen set V containing x and a countable set C such that V − C ⊆ B. Follows (V − C) ∩ A ⊆ B ∩ A and so (V ∩ A) − C ⊆ B ∩ A. Now x ∈ V, x ∈ (i, j)-SCl(A) such that V ∩ A 6= ∅ where V ∩ A is a (i, j)-ω-semiopen set, since V is a (i, j)-semiopen set and A ∈ τ1 ∩ τ2. Using the hypothesis, each nonempty (i, j)-ω-semiopen set of X is uncountable and so is (V ∩A)\C. Thus B∩A is uncountable. Therefore, B ∩ A 6= ∅ implies that x ∈ (i, j)-ω-SCl(A). Theorem 2.16. Let X be a bitopological space. If every (i, j)-ω-semiopen subset of X is τi-open in X. Then (X, (i, j)-ω-SO(X)) is a topological space. Proof. 1. ∅, X belong to (i, j)-ω-SO(X) 2. Let U, V ∈ (i, j)-ω-SO(X) and x ∈ U ∩ V. Then there exists (i, j)-semi open sets G, H in X containing x such that G\U and H\V are countable. Since (G ∩ H)\(U ∩ V) = (G ∩ H) ∩ ((X\U) ∪ (X\V)) ⊆ (G ∩ (X\U)) ∪ (H ∩ (X\V)) implies that (G ∩ H)\(U ∩ V) is a countable set and by hypothesis, the intersection of two (i, j)-semi open set is (i, j)-semi open. Hence U ∩ V ∈ (i, j)-ω- SO(X)). 3. The union follows directly. 3 (i, j)-ω-semicontinuous functions Definition 3.1. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be a (i, j)-ω-semicontinuous, if the inverse image of every σi-open set of Y is (i, j)-ω-semiopen in (X, τ1, τ2), where i 6= j, i, j=1, 2. Definition 3.2. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be a (i, j)-semicontinuous, if the inverse image of every σi-open set of Y is (i, j)-semiopen in (X, τ1, τ2), where i 6= j, i, j=1, 2. Theorem 3.3. Every (i, j)-semicontinuous function is (i, j)-ω-semicontinuous. 48 Carlos Carpintero, Sabir Hussain & Ennis Rosas CUBO 17, 3 (2015) Proof. The proof follows from the the fact that every (i, j)-semiopen set is (i, j)-ω-semiopen. However, the converse may be false. Example 3.4. Let X = {a, b, c}, τ1 = {∅, {a}, {b}, {a, b}, X}, τ2 = {∅, {a}, X}, σ1 = {∅, {a, b}, X}, σ2 = {∅, {a, c}, X}. Then the identity function f : (X, τ1, τ2) → (X, σ1, σ2) is (i, j)-ω-semicontinuous but not (i, j)-semicontinuous. Theorem 3.5. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following statements are equivalent: (1) f is (i, j)-ω-semicontinuous; (2) For each point x ∈ X and each σi-open set F in Y such that f(x) ∈ F, there is a (i, j)-ω- semiopen set A in X such that x ∈ A, and f(A) ⊂ F; (3) The inverse image of each σi-closed set in Y is a (i, j)-ω-semiclosed in X; (4) For A ⊆ X, f((i, j)-ω-SCl(A)) ⊂ σi-cl(f(A)); (5) For B ⊆ Y, (i, j)-ω-SCl(f−1(B)) ⊂ f−1(σi-cl(B)); (6) For C ⊆ Y, f−1(σi-Int(C)) ⊂ (i, j)-ω-SInt(f −1(C)). Proof. - (1)⇒(2): Let x ∈ X and F be a σi-open set of Y containing f(x). By (1), f −1(F) is (i, j)- ω-semiopen in X. Let A = f−1(F). Then x ∈ A and f(A) ⊂ F. (2)⇒(1): Let F be σi-open in Y and let x ∈ f −1(F). Then f(x) ∈ F. By (2), there is a (i, j)-ω- semiopen set Ux in X such that x ∈ Ux and f(Ux) ⊂ F implies x ∈ Ux ⊂ f −1(F). Hence f−1(F) is a (i, j)-ω-semiopen in X. (1)⇔(3): This follows from the fact that for any subset B of Y, f−1(Y\B) = X\f−1(B). (3)⇒(4): Let A ⊆ X. Since A ⊂ f−1(f(A)), we have A ⊂ f−1(σi-Cl(f(A))). By hypothesis f −1(σi- Cl(f(A))) is a (i, j)-ω-semiclosed set in X and hence (i, j)-ω-SCl(A)) ⊂ f−1(σi-Cl(f(A))). Follows f((i, j)-ω-SCl(A))) ⊂ f(f−1(σi-Cl(f(A))) ⊆ σi-Cl(f(A)). (4)⇒(3): Let F be any σi-closed subset of Y. Then f((i, j)-ω-SCl(f −1(F)) ⊂ σi-cl(f(f −1(F))) ⊂ σi- cl(F) = F. Therefore, the (i, j)-ω-SCl(f−1(F)) ⊂ f−1(F). Consequently, f−1(F) is a (i, j)-ω- semiclosed set in X. (4)⇒(5): Let B ⊆ Y. Now, f((i, j)-ω-SCl(f−1(B))) ⊂ σi-Cl(f(f −1(B))) ⊂ σi-Cl(B). Conse- quently, (i, j)-ω-SCl(f−1(B)) ⊂ f−1(σi-Cl(B)). (5)⇒(4): Let B = f(A) where A ⊆ X. Then, (i, j)-ω-SCl(A) ⊂ (i, j)-ω-SCl(f−1(B)) ⊂ f−1(σi- Cl(B)) = f−1(σi-Cl(f(A))), and hence f((i, j)-ω-SCl(A)) ⊂ σi-Cl(f(A)). (1)⇒(6): Let B ⊆ Y. Clearly, f−1(σi-Int(B)) is a (i, j)-ω-semiopen and we have f −1(σi-Int(B)) ⊂ (i, j)-ω-SInt(f−1σi-Int(B)) ⊂ (i, j)-ω-SInt(f −1B). (6)⇒(1): Let B be a σi-open set in Y. Then σi-Int(B) = B and f −1(B) ⊂ f−1(σi-Int(B)) ⊂ (i, j)- ω-SInt(f−1(B)). Hence, we have f−1(B) = (i, j)-ω-SInt(f−1(B)). This implies that f−1(B) is a (i, j)-ω-semiopen in X. CUBO 17, 3 (2015) (i, j)-ω-semiopen sets and (i, j)-ω-semicontinuity . . . 49 Definition 3.6. The graph G(f) of f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be (i, j)-ω-semiclosed in X × Y, if for each (x, y) ∈ (X× Y) \ G(f), there exists U ∈ (i, j)-ω-SO(X, x), i, j = {1, 2} with i 6= j and a σi-open set V of Y containing y such that (U × V) ∩ G(f) = ∅. Lemma 3.7. The graph G(f) of f : (X, τ1, τ2) → (Y, σ1, σ2) is (i, j)-ω-semiclosed in X × Y if and only if for each (x, y) ∈ (X × Y) \ G(f), there exists U ∈ (i, j)-ω-SO(X, x), i, j = {1, 2} with i 6= j and a σi-open set V of Y containing y such that f(U) ∩ V = ∅. Proof. The proof is an immediate consequence of Definition 3.6. Theorem 3.8. If a function f : (X, τ1, τ2) → (Y, σ1, σ2) is a (i, j)-ω-semicontinuous function and (Y, σi) is T1 i = {1, 2}, then G(f) is (i, j)-ω-semiclosed. Proof. Let (x, y) ∈ (X × Y) \ G(f). Then y 6= f(x). Since (Y, σi) is T1, there exist a σi-open set V and W of Y such that f(x) ∈ V and y /∈ W and V ∩ W = ∅. Since f is (i, j)-ω-semicontinuous, there exists U ∈ (i, j)-ω-SO(X, x) such that f(U) ⊂ V. Therefore, f(U) ∩W = ∅. Therefore, by Lemma 3.7, G(f) is (i, j)-ω-semiclosed. Definition 3.9. A bitopological space X is said to be a (i, j)-ω-semi-T2 space, if for each pair of distinct points x, y ∈ X, there exist U, V ∈ (i, j)-ω-SO(X) containing x and y, respectively, such that U ∩ V = ∅. Theorem 3.10. If f : (X, τ1, τ2) → (Y, σ1, σ2) is a (i, j)-ω-semicontinuous injective function and (Y, σi) is a T2 space, then (X, τ1, τ2) is a ω-semi-T2 space. Proof. The proof follows from the definition. Theorem 3.11. If f : (X, τ1, τ2) → (Y, σ1, σ2) is an injective (i, j)-ω-semicontinuous function with a (i, j)-ω-semiclosed graph, then X is a (i, j)-ω-semi-T2 space. Proof. Let x1 and x2 be any pair of distinct points of X. Then f(x1) 6= f(x2), so (x1, f(x2)) ∈ (X × Y)\G(f). Since the graph G(f) is (i, j)-ω-semiclosed, there exist a (i, j)-ω-semiopen set U containing x1 and V ∈ σi containing f(x2) such that f(U)∩V = ∅. Since f is (i, j)-ω-semicontinuous, f−1(V) is a (i, j)-ω-semiopen set containing x2 such that U∩f −1(V) = ∅. Hence X is (i, j)-ω-semi- T2. Definition 3.12. A collection {Uα : α ∈ I} of (i, j)-semiopen sets in a bitopological space X is called a (i, j)-semiopen cover of a subset A of X, if A ⊆ ⋃ α∈I Uα. Definition 3.13. A bitopological space X is said to be (i, j)-semi Lindeloff, if every (i, j)-semi open cover of X has a countable subcover. A subset A of bitopological space X is said to be (i, j)-semi Lindeloff relative to X, if every cover of A by (i, j)-semiopen sets of X has a countable subcover. Theorem 3.14. If X is a bitopological space such that every (i, j)-semiopen subset is (i, j)-semi Lindeloff relative to X. Then every subset is (i, j)-semi Lindeloff relative to X 50 Carlos Carpintero, Sabir Hussain & Ennis Rosas CUBO 17, 3 (2015) Theorem 3.15. For a bitopological space X. The following properties are equivalent: (1) X is (i, j)-semi Lindeloff. (2) Every countable cover of X by (i, j)-semiopen sets has a countable subcover. Proof. (2)⇒(1): Since every (i, j)-semiopen set is (i, j)-ω-semiopen, the proof follows. (1)⇒(2): Let {Uα : α ∈ I} be any cover of X by (i, j)-ω-semiopen sets of X. For each x ∈ X, there exists an αx ∈ I such that x ∈ Uαx. Since Uαx is a (i, j)-ω-semiopen, then there exists a (i, j)-semiopen set Vαx such that x ∈ Vαx and Vαx −Uαx is countable. The family {Vα : α ∈ I} is a (i, j)-semiopen cover of X and X is (i, j)-semi Lindeloff. Therefore there exists a countable subcover αxi with i ∈ N such that X = ⋃ i∈N Vαx i . Since X = ⋃ i∈N [(Vαx i − Uαx i ) ∪ Uαx i ] = ⋃ i∈N [(Vαx i − Uαx i ) ⋃ i∈N Uαx i ]. Since Vαx i − Uαx i is a countable set, for each α(xi), there exists a countable subset Iα(xi) of I such that Vαx i − Uαx i ⊆ ⋃ Iα(x i ) Uα and therefore X ⊆ ⋃ i∈N ( ⋃ α∈Iα(x i ) Uα) ∪ ( ⋃ i∈N Uα(xi)). Definition 3.16. A bitopological space X is called pairwise Lindeloff if each pairwise open cover of X has a countable subcover. Theorem 3.17. Let f : (X, τ1, τ2) → (Y, σ1, σ2) be a (i, j)-ω-semicontinuous function. If X is (i, j)-semi Lindeloff, then Y is pairwise Lindeloff. Proof. Let {Uα : α ∈ I} be any cover of Y by σi-open sets. Then {f −1(Uα) : α ∈ I} is a (i, j)-ω- semiopen cover of X. Since X is (i, j)-semi Lindeloff, there exists a countable subset I0 of I such that X = ⋃ α∈I0 Uα. Therefore, Y is a pairwise Lindeloff. Definition 3.18. A function f : (X, τ1, τ2) → (Y, σ1, σ2) is said to be: 1 (i, j)-ω-semiopen if f(U) is a (i, j)-ω-semiopen set in Y for every τi-open set U of X. 2 (i, j)-ω-semiclosed if f(U) is a (i, j)-ω-semiclosed set in Y for every τi-closed set U of X. Theorem 3.19. For a function f : (X, τ1, τ2) → (Y, σ1, σ2), the following properties are equivalent: (1) f is a (i, j)-ω-semiopen. (2) f(τi − Int(U)) ⊆ (i, j)-ω-SCl(f(U)), for each subset U of X. (3) τi − Int(f −1(V)) ⊆ f−1((i, j)-ω-SInt(V), for each subset V of Y. Proof. (1)⇒(2): Let U be any subset of X. Then τi − Int(U) is a τi-open set of X. Then f(τi − Int(U)) is a (i, j)-ω-semiopen set of Y. Since f(τi − Int(U)) ⊆ f(U), f(τi − Int(U)) = (i, j)- ω-SInt(f(τi − Int(U))) ⊆ (i, j)-ω-SInt(f(U)). (2)⇒(3):Let V be any subset of Y. Then f(τi − Int(f −1(V))) ⊆ (i, j)-ω-SInt(f(f−1(V))). Hence τi − Int(f −1(V)) ⊆ f−1((i, j)-ω-SInt(V)). (3)⇒(1): Let U be any τi-open set of X. Then τi − Int(U) = U. Now, V = τi − Int(V) ⊆ τi − CUBO 17, 3 (2015) (i, j)-ω-semiopen sets and (i, j)-ω-semicontinuity . . . 51 Int(f−1(f(V)) ⊆ f−1((i, j)-ω-SInt(f(V)))). Which implies that f(V) ⊆ f(f−1((i, j)-ω-SInt(f(V)))) ⊆ (i, j)-ω-SInt(f(V)). Hence f(V) is a (i, j)-ω-semiopen set of Y. Thus f is (i, j)-ω-semiopen. Theorem 3.20. Let f : (X, τ1, τ2) → (Y, σ1, σ2) be a function, then f is a (i, j)-ω-semiclosed function if and only if for each subset V of X, the (i, j)-ω-SCl(f(V)) ⊆ f(τi − Cl(V))). Proof. Let f be a (i, j)-ω-semiclosed function and V be any subset of X. Then f(V) ⊆ f(τi −Cl(V)) and f(τi − Cl(V)) is a (i, j)-ω-semiclosed set of Y. Hence (i, j)-ω-SCl(f(V)) ⊆ (i, j)-ω-SCl(f(τi − Cl(V))) = f(τi − Cl(V))). Conversely, let V be a τi-closed set of X. Then f(V) ⊆ (i, j)-ω- SCl(f(V)) ⊆ f(τi − Cl(V))) = f(V). Hence f(V) is a (i, j)-ω-semiclosed set of Y. Therefore, f is a (i, j)-ω-semiclosed function Definition 3.21. A bitopological space X is said to be (i, j)-ω-semiconnected, if X cannot be expressed as the union of two nonempty disjoint (i, j)-ω-semiopen sets. Definition 3.22. A bitopological space X is said to be pairwise connected [5], if it cannot be expressed as the union of two nonempty disjoint sets U and V such that U is τi-open and V is τj-open, where i, j = {1, 2} and i 6= j. Theorem 3.23. A (i, j)-ω-semicontinuous image of a (i, j)-ω-semiconnected space is pairwise connected. Proof. The proof is clear. Received: March 2015. Accepted: May 2015. References [1] S. Bose, Semi-open sets, Semi-Continuity and semi-open mappings in bitopological spaces, Bull. Calcutta Math. Soc., 73(1981), 237-246. [2] H. Z. Hdeib, ω-closed mappings, Revista Colombiana Mat., 16(1982), 65-78. [3] J. C. Kelly, Bitopological spaces, Proc. London Math. Soc., 13, pp. 71-89, (1963). [4] H. Maki, R. Chandrasekhara Rao and A. Nagoor Gani, On generalizing semi-open sets and preopen sets, Pure Appl. Math. Math. Sci, 49 (1999), pp 17-29. [5] W. J. Pervin, Connectedness in Bitopological spaces, Ind. Math., 29 (1967), 369-372. Introduction and Preliminaries (i, j)–semiopen sets (i,j)–semicontinuous functions