OPERATION-SEPARATION AXIOMS IN BITOPOLOGICAL SPACES

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1 An. Şt. Univ. Ovidius Constanţa Vol. 17(2), 2009, 5 18 OPERATION-SEPARATION AXIOMS IN BITOPOLOGICAL SPACES S.M. Al-Areefi Abstract In this paper, the concept of pairwise γ T 0, weak pairwise γ T 1, γ i T 1, pairwise γ T 1, weak pairwise γ T 2, γ i T 2, pairwise γ T 2, strong pairwise γ T 2, pairwise γ R 0 and pairwise γ R 1 are introduced and studied as a unification of several characterizations and properties of T 0, T 1, T 2, R 0 and R 1 in bitopological spaces 1. Introduction As a continuation of the study of operations in bitopological spaces introduced in [8], the aim of this paper is to introduce and study some separation axioms in bitopological spaces using the concept of operations on such spaces. In Section 2, we give a brief account of the definitions and results obtained in [8] which we need in this article. Section 3 is devoted to introduce the concepts of pairwise γ T 0, weak pairwise γ T 1,γ i T 1, pairwise γ T 1, weak pairwise γ T 2,γ i T 2, pairwise γ T 2 and strong pairwise γ T 2 and study some of their properties. Finally, the concepts of pairwise γ R 0 and pairwise γ R 1 are introduced and studied in Section 4. Throughout the paper, (X,τ 1,τ 2 ) and (Y,σ 1,σ 2 ) (or briefly, X and Y ) always mean bitopological spaces on which no separation axioms are assumed unless explicitly stated. Let A be a subset of X, by i int(a) and i cl(a), we denote respectively the interior and the closure of A with respect to τ i (or σ i ) for i = 1,2. Also i,j = 1,2 and i j. By id we mean the identity and nbd is the abbreviation of neighborhood. Also X \ A = A c is the complement of A in X. Key Words: Bitopological spaces, operation and separation axioms Mathematics Subject Classification: 54C55, 54D10, 54D15 Received: March, 2009 Accepted: September,

2 6 S.M. Al-Areefi 2. Preliminary Here, we give a brief account of the definitions and results obtained in [8] which we need in this paper. Definition 2.1. Let (X,τ 1,τ 2 ) be a bitopological space. An operation γ on τ 1 τ 2 is a mapping γ : τ 1 τ 2 P(X) such that V V γ for each V τ 1 τ 2, where V γ denotes the value of γ at V. The operators γ(v ) = V,γ(V ) = j cl(v ) and γ(v ) = i int(j cl(v )) for V τ i are operations in τ 1 τ 2. Definition 2.2. A subset A of a bitopological space (X,τ 1,τ 2 ) will be called a γ i -open set if for each x A, there exists a τ j - open set U such that x U and U γ A. τ iγ will denote the set of all γ i -open sets. Clearly we have τ iγ τ i. A subset B of X is said to be γ i -closed if X \ B is γ i -open in X. If γ(u) = u (resp. j cl(u)) and j cl(i int(u))) for each u T i then, the concept of γ i -open sets coincides with the concept of τ i -open (resp. ij o open [3] and cj δ open [3] sets. Definition 2.3. An operation γ on τ 1 τ 2 in (X,τ 1,τ 2 ) is said to be i-regular if for every pair U,V of τ i -open nbds of each point x X, there exists a τ i -open nbd W of x such that W γ U γ V γ. If γ is 1-regular and 2-regular then it is called pairwise regular or, simply, regular. Definition 2.4. An operation γ on (X,τ 1,τ 2 ) is said to be i-open if for every τ i -open neighborhood U of each x X, there exists a γ i -open set S such that x S and S U γ. If γ is 1-open and 2-open then it is called pairwise open or, simply, open. Example 2.5. Let X = {a,b,c} and let τ 1 = τ 2 = {φ, {a}, {b}, {a,b},x}. Let γ : τ 1 τ 2 P(X) be an operation defined by γ(b) = i cl(b) and λ : τ 1 τ 2 P(X) be an operation defined by λ(b) = i int(j cl(b)). Then we have τ iγ = {φ,x} and τ 1λ = τ i. It is easy to see that γ is i-regular but it is not i-open on (X,τ 1,τ 2 ). Example 2.6. Let X = {a,b,c} and let τ 1 = τ 2 = {φ,x, {a}, {b}, {a,b}, {a,c}}. For b X we define an operation γ : τ 1 τ 2 P(X) by { γ(a) = A γ A if b A = i cl(a) if b A Then the operation γ : τ 1 τ 2 P(X) is not regular on τ 1 τ 2. In fact, let U = {a} and V = {a,b} be i-open neighborhoods of a, then U γ V γ = {a,c} {a,c} = {a} and W γ U γ V γ for any i-open neighborhood W of a.

3 OPERATION-SEPARATION AXIOMS IN BITOPOLOGICAL SPACES 7 Definition 2.7. Let γ,µ be two operations on τ 1 τ 2 in (X,τ 1,τ 2 ). Then (X,τ 1,τ 2 ) is called (γ,µ) i -regular if for each x X and each τ i -open nbd V of x, there exists a τ i -open set U containing x such that U γ V µ. If µ = id, then the (γ,µ) i -regular space is called γ i -regular. If (X,τ 1,τ 2 ) is (γ,µ) 1 -regular and (γ,µ) 2 -regular, then it is called pairwise (γ,µ)-regular. Moreover, we can show that this operation is i-open on τ 1 τ 2. In the rest of this paper, instead of γ operation on τ 1 τ 2 in (X,τ 1,τ 2 ), we shall say γ operation on (X,τ 1,τ 2 ). Proposition 2.8. Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then (X,τ 1,τ 2 ) is a γ i -regular space if and only if τ i = τ iγ holds. Proposition 2.9. Let γ be an i-regular operation on a bitopological space (X,τ 1,τ 2 ), then (i) If A and B are γ i -open sets of X, then A B is γ i -open. (ii) τ iγ is a topology on X. Remark If γ is not i-regular, then the above proposition is not true in general. In Exmaple 2.6, γ is not i-regular. In fact τ iγ = {φ,x, {b}, {a,b}{a,c}} which is not a topology on X. Definition Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). A point x X is in the γ i -closure of a set A X if U γ A φ for each τ i -open nbd U of x. The γ i -closure of the set A is denoted by cl iγ (A). A subset A of X is said to be γ i -closed (in the sense of cl iγ (A)) if cl iγ (A) A. Definition Let A (X,τ 1,τ 2 ). For the family τ iγ, we define a set τ iγ cl(a) as follows: τ iγ cl(a) = {F : A F,X \ F τ iγ }. Proposition For a point x X, x τ iγ cl(a) if and only if V A φ for any V τ iγ such that x V. Remark It is easily shown that for any subset A of (X,τ 1,τ 2 ), A τ i cl(a) cl iγ (A) τ iγ cl(a). Theorem Let γ be an operation on a bitopological space (X,τ 1,τ 2 ), then (i) cl iγ (A) is τ i -closed in (X,τ 1,τ 2 ). (ii) If (X,τ 1,τ 2 ) is γ i -regular, then cl iγ (A) = τ i cl(a) holds.

4 8 S.M. Al-Areefi (iii) If γ is open, then cl iγ (A) = τ iγ cl(a) and cl iγ (cl iγ (A)) = cl iγ (A) hold and cl iγ (A) is γ i -closed (in the sense of Definition 2.11). Theorem Let γ be an operation on a bitopological space (X,τ 1,τ 2 ) and A X, then the following are equivalent: (a) A is γ i -open in (X,τ 1,τ 2 ). (b) cl iγ (X \ A) = X \ A (i.e., X \ A is γ i -closed in the sense of Definition 2.11). (c) τ iγ cl(x \ A) = X \ A. (d) X \ A is γ i -closed (in the sense of Definition 2.12) in (X,τ 1,τ 2 ). Lemma If γ is a regular operation on (X,τ 1,τ 2 ), then cl iγ (A B) = cl iγ (A) cl iγ (B), for any subsets A and B of X. Corollary If γ is a regular and open operation on (X,τ 1,τ 2 ), then cl iγ satisfies the Kuratowski closure axioms. Definition A mapping f : (X,τ 1,τ 2 ) (Y,σ 1,σ 2 ) is said to be (γ,β) i - continuous if for each point x of X and each σ i -open set V containing f(x), there exists a τ i -open set U such that x U and f(u γ ) V β. If f is (γ,β) i - continuous for i = 1,2, then it is called pairwise (γ,β)-continuous. Example If (γ,β) = (id,id) (resp. id,j cl),(j cl,id),(j cl j cl),(id, i int j cl),(i int j cl, id),(i int j cl, j cl),(j cl, i int j cl) and (i int j cl,i int j cl)) then pairwise (γ,β)-continuity coincides with pairwise continuity [7] (resp. pairwise weak continuity [5], pairwise strong θ-continuity [3], pairwise θ-continuity [4], pairwise almost continuity [5], pairwise super continuity [9], pairwise weak θ-continuity [11], pairwise almost strong θ-continuity [3] and pairwise δ-continuity [3]). Proposition Let f : (X,τ 1,τ 2 ) (Y,σ 1,σ 2 ) be a (γ,β) i -continuous mapping, then (i) f(cl iγ (A)) cl iβ (f(a)) for every A X. (ii) for any β i -closed set B of Y, f 1 (B) is γ i -closed in X, i.e., for any U σ iβ, f 1 (U) τ iγ. Definition Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). A subset K X is said to be γ i -compact if for every τ i -open cover C of K, n there exists a finite subfamily {G 1,...,G n } of C such that K G γ r. r=1

5 OPERATION-SEPARATION AXIOMS IN BITOPOLOGICAL SPACES 9 Example If γ = id (resp. j cl and i int j cl) then γ i compactness coincides with τ i -compactness (resp. ij-almost-compactness [11] and ij-nearcompactness [3]). Theorem Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). If X is γ i -compact, then every cover of X by γ i -open sets has a finite subcover. If γ is open, then the converse is true. 3. Pairwise γ T k Spaces Definition Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is called pairwise γ T 0 if for any two distinct points of X, there exists a subset which is either γ i -open or γ j -open containing one of the points but not the other. Example In Definition 3.25, if γ = id, then we obtain the definition of pairwise T 0 spaces [6]. If γ(u) = j cl(u) (resp. i int(j cl(u)) for U τ i, then pairwise γ T 0 spaces are called pairwise θ T 0 (resp. pairwise δ T 0 ). If γ(u) = i int(i cl(u)) for U τ i, then we obtain pairwise rt 0 spaces [2]. Theorem Let γ be an operation on a bitopological space (X.τ 1,τ 2 ). If X is pairwise γ T 0, then for any two distinct points x and y of X, there exists a subset U which is either τ i -open or τ j -open containing one of them (say x) such that y U γ. If γ is open, then the converse is true. Proof. Let x,y X such that x y, then there exists a set U which is (say) τ iγ -open such that x U and y U. Then there exists a τ i -open set G containing x such that G γ U. Obviously, y G γ. Now, if γ is open, let x,y X such that x y. Then, by assumption, there exists a τ i -open set U such that x U and y U γ. Since γ is open, there exists a τ iγ -open set S such that x S and S U γ. Obviously, y S. This shows that X is pairwise γ T 0. Theorem Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is pairwise γ T 0 if and only if for each two distinct points x and y of X, either τ iγ cl{x} τ iγ cl{y} or τ jγ cl{x} τ jγ cl{y}. Proof. Let X be a pairwise γ T 0 space and x,y X such that x y. Suppose U is a γ i -open set containing x but not y. Then y τ iγ cl{y} X \U and so x τ iγ cl{y}. Hence τ iγ cl{x} τ iγ cl{y}. Conversely, let x,y X such that x y. Then either τ iγ cl{x} τ iγ cl{y} or τ jγ cl{x} τ jγ cl{y}. In the former case, let z X such that z τ iγ cl{y} and z τ iγ cl{x}. We

6 10 S.M. Al-Areefi assert that y τ iy cl{x}. If y τ iγ cl{x}, then τ iγ cl{y} τ iγ cl{x}, so z τ iγ cl{y} τ iγ cl{x}, a contradiction. Hence y τ iγ cl{x} and therefore U = X \ τ iγ cl{x} is a γ i -open set containing y but not x. The case τ jγ cl{x} τ jγ cl{y} can be dealt with similarly. Corollary A bitopological space (X,τ 1,τ 2 ) is pairwise θ T 0 if and only if for each two distinct points x and y of X, either ij cl θ {x} ij cl θ {y} or ji cl θ {x} ji cl θ {y}. Corollary A bitopological space (X,τ 1,τ 2 ) is pairwise δ T 0 if and only if each two distinct points x and y of X, either ij cl δ {x} ij cl δ {y} or ji cl δ {x} ji cl δ {y}. Definition Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is called weakly pairwise γ T 1 if for any two distinct points x and y of X, there exists a γ i -open set U and a γ j -open set V such that either x U \ V and y V \ U or y U \ V and x V \ U. Example In Definition 3.31, if γ = id, then we obtain the definition of weakly pairwise T 1 [10]. If γ(u) = j cl(u) (resp. i int(j cl(u))) for U τ i, then weakly pairwise γ T 1 spaces are called weakly pairwise θ T 1 (resp. weakly pairwise δ T 1 ). If γ(u) = i int(i cl(u)) for U τ i, then we obtain weakly pairwise rt 1 spaces [2]. Theorem Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then the following are equivalent: (i) (X,τ 1,τ 2 ) is weakly pairwise γ T 1. (ii) τ 1γ cl{x} τ 2γ cl{x} = {x} for every x X. (iii) For every x X, the intersection of all γ 1 -open nbds and all γ 2 -open nbds of x is {x}. Proof. (i) (ii): Let x X and y τ 1γ cl{x} τ 2γ cl{x}, where y x. Since X is weakly pairwise γ T 1, there exists a γ 1 -open set U such that y U, x U or there exists a γ 2 -open set V such that y V, x V. In either case, y τ 1γ cl{x} τ 2γ cl{x}. Hence {x} = τ 1γ cl{x} τ 2γ cl{x}. (ii) (iii): If x,y X such that x y, then x τ 1γ cl{y} τ 2γ cl{y}, so there is a γ 1 -open set or a γ 2 -open set containing x but not y. Therefore y does not belong to the intersection of all γ 1 -open nbds and all γ 2 -open nbds of x. (iii) (i): Let x and y be two distinct points of X. By (iii), y does not belong to a γ 1 -nbd or a γ 2 -nbd of x. Therefore there exists a γ 1 -open or a γ 2 -open set containing x but not y. Hence x is a weakly pairwise γ T 1 space.

7 OPERATION-SEPARATION AXIOMS IN BITOPOLOGICAL SPACES 11 Corollary For a bitopological space (X,τ 1,τ 2 ), the following are equivalent: (i) (X,τ 1,τ 2 ) is weakly pairwise θ T 1. (ii) 12 cl θ {x} 21 cl θ {x} = {x} for every x X. (iii) For every x X, the intersection of all 12 θ-open nbds and all 21 θ- open nbds of x is {x}. Corollary For x X, the intersection of all 12 δ-open neighborhoods and all 21 θ-open neighborhoods of x is {x}. Theorem Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). If X is weakly pairwise γ T 1, then for any two distinct points x and y of X, there exists a τ i -open set U and a τ j -open set V such that either x U, y U γ and y V, = x V γ or y U, x U γ and x V,y V γ. If γ is open, then the converse is true. Proof. Similar to that of Theorem Definition Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is called γ i T 1 if for any two distinct points x and y of X, there exist two γ i -open sets U and V such that x U \ V and y V \ U. If X is γ 1 T 1 and γ 2 T 1, then it is called pairwise γ T 1. Example In Definition 3.37, if γ = id, then we obtain the definition of pairwise T 1 [14]. If γ(u) = j cl(u) (resp. i int(j cl(u))) for U τ i, then pairwise γ T 1 spaces are called pairwise θ T 1 (resp. pairwise δ T 1 ) and γ i T 1 spaces are called ij θ T 1 (resp. ij δ T 1 ) spaces. Theorem Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is γ i T 1 if and only if τ iγ cl{x} = {x}, for every x X. Proof. Let y {x}, then y x and there exists a τ iγ -open set U such that y U and x U. Therefore y τ iγ cl{x} and so τ iγ cl{x} = {x}. Conversely, let x,y X such that x y. Since τ iγ cl{x} = {x} and τ iγ cl{y} = {y}, then there exists a γ i -open set U and γ i -open set V such that x U \ V and y V \ U. Thus X is γ i T 1. Corollary Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is pairwise γ T 1 if and only if τ 1γ cl{x} = {x} = τ 2γ cl{x}, for every x X. Corollary A bitopological space (X,τ 1,τ 2 ) is pairwise T 1 if and only if τ 1 cl{x} = {x} = τ 2 cl{x}, for every x X.

8 12 S.M. Al-Areefi Corollary A bitopological space (X,τ 1,τ 2 ) is pairwise θ T 1 if and only if 12 cl θ {x} = {x} = 21 cl θ {x}, for every x X. Corollary A bitopological space (X,τ 1,τ 2 ) is pairwise δ T 1 if and only if 12 cl δ {x} = {x} = 21 cl δ {x}, for every x X. Definition Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is called weakly pairwise γ T 2 if for any two distinct points x and y of X, there exist a γ i -open set U and a disjoint γ j -open set V such that x U and y V or x V and y U. Example If γ = id in Definition 3.44, we obtain the definition weakly pairwise T 2 [15]. In case γ(u) = j cl(u) (resp. i int(j cl(u))) for U τ i, then weakly pairwise γ T 2 is called weakly pairwise θ T 2 (resp. weakly pairwise δ T 2 ). If γ(u) = i int(i cl(u)) for U τ i, then we obtain a pairwise semi rt 2 [2]. Definition Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is called γ i T 2 if for any two distinct points x and y of X, there exist two disjoint γ i -open sets U and V such that x U and y V. If X is γ 1 T 2 and γ 2 T 2, then it is called pairwise γ T 2. Definition Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is called strongly pairwise γ T 2 if for any two distinct points x and y of X, there exist a γ i -open set U and a disjoint γ j -open set V such that x U and y V. Example If γ = id in Definition 3.47, we obtain the definition pairwise T 2 [7]. In case γ(u) = j cl(u) (resp. i int(j cl(u))) for U τ i, then strongly pairwise γ T 2 is called strongly pairwise θ T 2 (resp. strongly pairwise δ T 2 ). If γ(u) = i int(i cl(u)) for U τ i, then we obtain pairwise rt 2 [2]. Theorem Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is strongly pairwise γ T 2 if and only if the intersection of all γ i - closed γ j -nbd of each point of X is reduced to that point. Proof. Let X be strongly pairwise γ T 2 and x X. To each y X, x y, there exist a γ i -open set G and a γ j -open set H such that x H, y G and G H = φ. Since x H X \ G, therefore X \ G is γ i -closed γ j -nbd of x to which y does not belong. Consequently, the intersection of all γ i -closed γ j -nbds of X is reduced to {x}. Conversely, let x,y X such that x y, then by hypothesis, there exists a γ i -closed γ j -nbd U of x such that y U. Now, there exists a γ i -open set G such that x G U. Thus G is a γ i -open set, X \ U is a γ j -open set, x G, y X \ U and G X \ U = φ. Hence X is strongly pairwise γ T 2.

9 OPERATION-SEPARATION AXIOMS IN BITOPOLOGICAL SPACES 13 Theorem Let γ be an operation on a bitopological space (X,τ 1,τ 2 ). Then X is γ i T 2 if and only if the intersection of all γ i -closed γ i -nbds of each point of X is reduced to that point. Proof. Similar to that of Theorem Theorem Let γ be a regular operation on a strongly pairwise γ T 2 space (X,τ 1,τ 2 ) and A X a γ i -compact. Then A is γ j -closed. Proof. If A = X, then A is obviously γ j -closed. If A X, then there is a point x X \ A. Since X is strongly pairwise γ T 2, for every y A, there exist a γ j -open set U y and a γ i -open set V y such that x U y, y V y and U y V y = φ. Then {V y : y A} is a γ i -open cover of A which is γ i -compact, n then there exists a finite subfamily V y1,...,v yn such that A V yr. Let U = n U yr and V = r=1 r=1 n V yr. Since γ is regular, therefore U is a γ j -open set, r=1 V is a γ i -open set, x U, A V and U U = φ. Thus x U X \ A and so X \ A is γ j -open and so A is γ j -closed. Corollary If A is an ij-almost-compact subset of a strongly pairwise θ T 2 space (X,τ 1,τ 2 ), then A is ji θ-closed. Corollary If A is an ij-nearly-compact subset of a strongly pairwise δ T 2 space (X,τ 1,τ 2 ), then A is ji δ-closed. Theorem Let γ (resp. β) be an operation on a bitopological space (X,τ 1,τ 2 ) (resp. (Y,σ 1,σ 2 )) and f : X Y be a pairwise (γ,β)-continuous injection. If Y is pairwise β T 0 (resp. weak pairwise β T 1,β i T 1, pairwise β T 1, weak pairwise β T 2,β i T 2, pairwise β T 2 and strong pairwise β T 2 ), then X is pairwise γ T 0 (resp. weak pairwise γ T 1,γ i T 1, pairwise γ T 1, weak pairwise γ T 2,γ i T 2, pairwise γ T 2 and strong pairwise γ T 2 ). Proof. Suppose that Y is strong pairwise β T 2 and let x,y X such that x y. Then there exist a σ iβ -open set U and a σ jβ -open set V such that f(x) U, f(y) V and U V = φ. Since f is pairwise (γ,β)-continuous, by Proposition 2.21, f 1 (U) is τ iγ -open and f 1 (V ) is τ jγ -open. Also, x f 1 (U),y f 1 (V ) and f 1 (U) f 1 (V ) = φ. This show that X is strong pairwise γ T 2. The proofs of the other cases are similar. For different choices for γ and β in Theorem 3.54, we can write down a lot of results, for example, we have the following

10 14 S.M. Al-Areefi Corollary If f : (X,τ 1,τ 2 ) (Y,σ 1,σ 2 ) is pairwise weakly continuous injection and Y is a strong pairwise θ T 2 space, then X is pairwise Hausdorff. Corollary If f : (X,τ 1,τ 2 ) (Y,σ 2,σ 2 ) is pairwise almost continuous injection and Y is a strong pairwise δ T 2 space, then X is pairwise Hausdorff. Corollary If f : (X,τ 1,τ 2 ) (Y,σ 2,σ 2 ) is pairwise strongly θ-continuous injection and Y is a pairwise Hausdorff space, then X is strong pairwise θ T 2. Corollary If f : (X,τ 1,τ 2 ) (Y,σ 2,σ 2 ) is pairwise super continuous injection and Y is a pairwise Hausdorff space, then X is strong pairwise δ T 2. Corollary If f : (X,τ 1,τ 2 ) (Y,σ 2,σ 2 ) is pairwise θ-continuous injection and Y is a strong pairwise θ T 2 space, then X is strong pairwise θ T 2. Corollary If f : (X,τ 1,τ 2 ) (Y,σ 2,σ 2 ) is pairwise weakly θ-continuous injection and Y is a strong pairwise θ T 2 space, then X is strong pairwise δ T 2. Definition Let γ be an operation on a bitopological space (X,τ 1,τ 2 ) and H X, then the relative operation γ : τ 1 /H τ 2 /H P(H) is an operation on (H,τ 1 /H,τ 2 /H) defined by (G H) γ = G γ H for each G τ 1 τ 2, where τ i /H = {U H : U τ i }. Lemma Let γ be an operation on a bitopological space (X,τ 1,τ 2 ), H X and γ be the relative operation in the subspace (H,τ 1 /H,τ 2 /H). If U is γ i -open in X, then U H is γi -open in H. Theorem Every subspace of a pairwise γ T 0 (resp. weak pairwise γ T 1,γ i T 1, pairwise γ T 1, weak pairwise γ T 2,γ i T 2, pairwise γ T 2 and strong pairwise γ T 2 ) is a pairwise γ T 0 (resp. weak pairwise γ T 1,γ i T 1, pairwise γ T 1, weak pairwise γ T 2,γ i T 2, pairwise γ T 2 and strong pairwise γ T 2 ). Proof. Follows directly from Lemma Remark The argument of Theorem 3.36 is true for the cases of weak pairwise γ T 1,γ i T 1, pairwise γ T 1, weak pairwise γ T 2,γ i T 2, pairwise γ T 2 and strong pairwise γ T 2 spaces. For different choices for γ in Remark 3.64, we can write down a lot of results, for example, we have the following

11 OPERATION-SEPARATION AXIOMS IN BITOPOLOGICAL SPACES 15 Corollary A bitopological space (X,τ 1,τ 2 ) is weak pairwise δ T 1 if and only if for any two distinct points x and y of X, there exist a τ i -open set U and a τ j -open set V such that either x U,y i int(j cl(u)) and y V,x j int(i cl(v )) or y U, x i int(j cl(u)) and x V,y j int(i cl(v )). Corollary A bitopological space (X,τ 1,τ 2 ) is weak pairwise δ T 2 if and only if for any two distinct points x and y of X, there exist a τ i -open set U and a τ j -open set V such that either x U and y V or y U, and x V and i int(j cl(u)) j int(i cl(v )) = φ. Corollary A bitopological space (X,τ 1,τ 2 ) is strong pairwise δ T 2 if and only if for any two distinct points x and y of X, there exist a τ i -open set U and a τ j -open set V such that x U,y V, and i int(j cl(u)) j int(i cl(v )) = φ. Theorem Let γ be an open and regular operation on a bitopological space (X,τ 1,τ 2 ). Then (X,τ 1,τ 2 ) is pairwise γ T 0 (resp. weak pairwise γ T 1, pairwise γ T 1, weak pairwise γ T 2 and strong pairwise γ T 2 ) if and only if (X,τ 1γ,τ 2γ ) is pairwise T 0 (resp. weak pairwise T 1, pairwise T 1, weak pairwise T 2, and strong pairwise T 2 ). Proof. It is straight forward by Proposition 2.9 and Remark 3.64 For different choices for γ in Theorem 3.68, we can write down a lot of results, for example, we have the following Corollary A bitopological space (X,τ 1,τ 2 ) is pairwise δ T 0 if and only if (X,τ 1δ,τ 2δ ) is pairwise T 0. Corollary A bitopological space (X,τ 1,τ 2 ) is pairwise δ T 1 if and only if (X,τ 1δ,τ 2δ ) is pairwise T 1. Corollary A bitopological space (X,τ 1,τ 2 ) is strong pairwise δ T 2 if and only if (X,τ 1δ,τ 2δ ) is strong pairwise T 2. We finish this section by investigating general operator approaches of closed graphs of mappings. Definition [13] Let (X,τ 1,τ 2 ) and (Y,σ, σ 2 ) be two bitopological spaces. The cross product of the two spaces X and Y is defined to be the space (X Y,µ 1,µ 2 ), where µ i = τ i σ j. Definition Let γ (resp. β) be an operation on the bitopological space (X,τ 1,τ 2 ) (resp. (Y,σ 1,σ 2 )). An operation ρ : τ 1 σ 2 τ 2 σ 1 P(X Y ) is

12 16 S.M. Al-Areefi said to be associated with γ and β if (U V ) ρ = U γ V β for each U τ i and V σ j. ρ is called regular with respect to γ and β if for each (x,y) X Y and each µ j -open nbd W of (x,y), there exist a τ i -open nbd U of x and a σ j -open nbd V of y such that W ρ U γ V β. Theorem Let ρ be an operation on X X associated with γ and γ. If f : X Y is a pairwise (γ,β)-continuous mapping and (Y,σ 1,σ 2 ) is a strong pairwise β T 2 space, then the set A = {(x,y) X X : f(x) = f(y)} is a ρ i -closed subset of X X. Proof. We show that cl iρ (A) A. Let (x,y) X X \ A. Then there exist a σ i -open set U and a σ j -open set V such that f(x) U,f(y) V and U β V β = φ. Moreover, there exist a τ i -open set W and a τ j -open set S such that x W,y S and f(w γ ) U β and f(s γ ) V β. Therefore, (W S) ρ A = φ. This shows that (x,y) cl iρ (A) and so cl iρ (A) A. Corollary Let ρ be an operation on X X associated with γ and γ which is regular with respect to γ and γ. A space X is strong pairwise γ T 2 if and only if the diagonal set = {(x,x) : x X} is ρ i -closed in X Y. Theorem Let ρ be an operation on X Y associated with γ and β. If f : X Y is a pairwise (γ,β)-continuous mapping and (Y,σ 1,σ 2 ) is a strong pairwise β T 2 space, then the graph of f, G(f) = {(x,f(x)) X Y } is a ρ i -closed subset of X Y. Proof. The proof is similar to that of Theorem 3.74 References [1] Abd El-Monsef, M.E. Kozae, A.M. and Taher, B.M., Compactification in bitopological spaces, Arab J. for Sci. Engin., 22(1A)(1997), [2] Arya, S.P. and Nour, T.M., Separation axioms for bitopological spaces, Indian J. Pure Appl. Math., 19(1)(1988), [3] Banerjee, G.K., On pairwise almost strongly θ-continuous mappings, Bull. Calcutta Math. Soc., 79(1987), [4] Bose, S. and Sinha, D., Almost open, almost closed, θ-continuous and almost quasi compact mappings in bitopological spaces, Bull. Calcutta Math. Soc., 73(1981), [5] Bose, S. and Sinha, D., Pairwise almost continuous map and weakly continuous map in bitopological spaces, Bull. Calcutta Math. Soc., 74(1982),

13 OPERATION-SEPARATION AXIOMS IN BITOPOLOGICAL SPACES 17 [6] Fletcher, P., Hoyle, H.B. III and Patty, C.W., The comparison of topologies, Duke Math. J., 36(2)(1969), [7] Kelly, J.C., Bitopological spaces, Proc. London Math. Soc., (3)13(1963), [8] Khedr, F.H., Operations on bitopological spaces (submitted). [9] Khedr, F.H. and Al-Shibani, A.M., On pairwise super continuous mappings in bitoplogical spaces, Internat. J. Math. Math. Sci., (4)14(1991), [10] Misra, D., N., and Dube, K.K., Pairwise R 0 -spaces, Ann de la Soc. Sci, Bruxells, T. 87(1)(1973), [11] Mukherjee, M.N., On pairwise almost compactness and pairwise H- closedness in a bitopologicla space, Ann. Soc. Sci. Bruxelles, T. 96, 2(1982), [12] Murdeshware, M.G. and Namipally, S.A., Quasi-uniform topological spaces, Monograph, P. Noordoff Ltd. (1966). [13] Nandi, J.N. and Mukherjee, M.N., Bitopological δ-cluster sts and some applications, Indiana J. Pure Appl. Math., 29(8)(1998), [14] Reilly, I.L., On bitopological separation properties, Nanta Math., 5(2)(1972), [15] Smithson, R.E., Mutlifunctions and bitopological spaces, J. Nat. Sci. Math., 11(2)(1971), [16] Swart, J., Total disconnectedness in bitopological spaces and product bitopological spaces, Proc. Kon. Ned. Akad Wetench A 74 = Indag. Math., 33(2)(1971), Faculty of Science Department of Mathematics P.O. Box 838, Dammam 31113, Kingdom of Saudi Arabia s.alareefi@yahoo.com

14 18 S.M. Al-Areefi