SOME FUNDAMENTAL PROPERTIES OF -OPEN SETS IN IDEAL BITOPOLOGICAL SPACES M. CALDAS, S. JAFARI AND N. RAJESH The aim of this paper is to introduced and characterized the concepts of -open sets and their related notions in ideal bitopological spaces. Abstract.
1. Introduction Kuratowski [9] and Vaidyanathasamy [11] introduced and investigated the concept of ideals in topological spaces . An ideal I on a topological space (X; ) is a nonempty collection of subsets of X which satis es (i) A 2 I and B A implies B 2 I and (ii) A 2 I and B 2 I implies A [ B 2 I . Given a bitopological space (X; 1; 2) with an ideal I on X and if P (X ) is the set of all subsets of X , a set operator (:)i : P (X ) ! P (X ), called the local function [11] of A with respect to i and I , is de ned as follows: for A X , Ai ( i ; I ) = fx 2 X jU \ A 2= I for every U 2 i (x)g, where i (x) = fU 2 i jx 2 U g. For every ideal topological space (X; ; I ), there exists a topology (I ), ner than , generated by the base (I ; ) = fU nI j U 2 and I 2 Ig, but in general (I ; ) is not always a topology [6]. Observe additionally that i -Cl (A) = A [ Ai ( i ; I ) de nes a Kuratowski closure operator for (I ), when there is no chance of confusion, Ai (I ) is denoted by Ai and i -Int (A) denotes the interior of A in i (I ). The aim of this paper is to introduced and characterized the concepts of -open sets and their related notions in ideal bitopological spaces. 2. Preiliminaries Let A be a subset of a bitopological space (X; 1 ; 2 ). We denote the closure of A and the interior of A with respect to i by i -Cl(A) and i -Int(A), respectively. De nition 2.1. A subset A of a bitopological space (X; 1 ; 2 ) is said to be (i; j )-semiopen [7] (resp. (i; j )-preopen [7], (i; j )-semi-preopen [8]) if A i -Cl( j -Int(A)) (resp. A i -Int( j -Cl(A)), A i -Cl( j -Int( i Cl(A)))), where i; j = 1; 2 and i 6= j . 2000 Mathematics Subject Classi cation. 54D10. Key words and phrases. Ideal bitopological spaces, (i; j )- -I -open sets, (i; j )- I -closed sets. 1
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De nition 2.2. A subset A of an ideal bitopological space (X; 1 ; 2 ; I ) is said to be
(i; j )-semi-I -open [3] if A j -Cl ( i -Int(A)). (i; j )-pre-I -open [2] if A i -Int( j -Cl (A)). (i; j )-b-I -open [4] if A i -Int( j -Cl (A)) [ j -Cl ( i -Int(A)) . (i; j )- -I -open [5] if A j -Int( i -Cl ( j -Int(A))). (i; j )-semi-precontinuous [8] if the inverse image of every j open set in (Y; 1 ; 2 ) is (i; j )-semi-preopen in (X; 1 ; 2 ; I ), where i 6= j , i; j =1, 2. De nition 2.3. A function f : (X; 1 ; 2 ; I ) ! (Y; 1 ; 2 ) is said to be (i) pairwise pre-I -continuous [2] if the inverse image of every i open set of Y is (i; j )-pre-I -open in X , where i 6= j , i; j =1, (i) (ii) (iii) (iv) (v)
(ii) (iii) (iv)
2. pairwise semi-I -continuous [3] if the inverse image of every i open set of Y is (i; j )-semi-I -open in X , where i 6= j , i; j =1, 2. pairwise b-I -continuous [4] if the inverse image of every i -open set of Y is (i; j )-b-I -open in X , where i 6= j , i; j =1, 2. pairwise -I -continuous [5] if the inverse image of every i open set of Y is (i; j )- -I -open in X , where i 6= j , i; j =1, 2.
3. (i; j )- -I -open sets
De nition 3.1. A subset A of an ideal bitopological space (X; 1 ; 2 ; I ) is said to be (i; j )- -I -open if and only if A j -Cl( i -Int( j -Cl (A))), where i; j = 1; 2 and i 6= j . The family of all (i; j )- -I -open sets of (X; 1 ; 2 ; I ) is denoted by I O(X; 1 ; 2 ) or (i; j )- I O(X ). Also, The family of all (i; j )- -I open sets of (X; 1 ; 2 ; I ) containing x is denoted by (i; j )- I O(X; x). Remark 3.2. Let I and J be two ideals on (X; 1 ; 2 ). If I J , then J O(X; 1 ; 2 ) I O(X; 1 ; 2 ). Proposition 3.3. (i) Every (i; j )-b-I -open set is (i; j )- -I -open. (ii) Every (i; j )- -I -open set is (i; j )-semi-preopen.
The proof follows from the de nitions. The following example show that the converses of Proposition 3.3 is not true in general. Example 3.4. Let X = fa; b; cg 1 = f;; fag; X g, 2 = f;; fag; fa; bg; X g and I = f;; fagg. Then the set fa; cg is (i; j )- -I -open but not (i; j )Proof.
b-I -open. Corollary 3.5. (i) Every (i; j )- -I -open set is (i; j )- -I -open. (ii) Every (i; j )-semi-I -open set is (i; j )- -I -open. (iii) Every (i; j )-pre-I -open set is (i; j )- -I -open.
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Proposition 3.6. For an ideal bitopological space (X; 1 ; 2 ; I ) and A X , we have: (i) If I = f;g, then A is (i; j )- -I -open if and only if A is (i; j )semi-preopen.
(ii) If I = P (X ), then A is (i; j )- -I -open if and only if A is (i; j )semiopen.
The proof follows from the fact that (i) If I = f;g, then A = Cl(A). (ii) If I = P (X ), then A = ; for every subset A of X .
Proof.
Remark 3.7. The intersection of two (i; j )- -I -open sets need not be (i; j )- -I -open as it can be seen from the following example. Example 3.8. Let X = fa; b; c; dg, 1 = f?, fag, fbg, fa; bg, fa; b; cg, X g, 2 = f?, X g and I = f?; fcg; fdg; fc; dgg. Then the sets fa; cg and fb; cg are (1; 2)- -I -open sets of (X; 1 ; 2 ; I ) but their intersection fcg is not an (1; 2)- -I -open set of (X; 1; 2; I ). Theorem 3.9. If fA g 2 is a family of (i; j )- -I -open sets in (X; 1 ; 2 ; I ), S then A is (i; j )- -I -open in (X; 1 ; 2 ; I ). 2
Since fA : 2 g (i; j )- I O(X ), then A j -Cl( i Int( j -Cl (A ))) for every 2 . Thus, [2 A [2 j -Cl( i -Int( j Cl (A ))) j -Cl( i -Int( [2 j -Cl (A ))) = j -Cl( i -Int( j -Cl ( [2 A ))). Therefore, we obtain [2 A j -Cl( i -Int( j -Cl ( [2 A ))). Hence any union of (i; j )- -I -open sets is (i; j )- -I -open. Theorem 3.10. A subset A of an ideal bitopological space (X; 1 ; 2 ; I ) is (i; j )- -I -open if and only if j -Cl(A) = j -Cl( i -Int( j -Cl (A))). Proof. Let A be an (i; j )- -I -open set of X . Then, we have A j Cl( i -Int( j -Cl (A))) and hence j -Cl(A) j -Cl( i -Int( j -Cl (A))) j -Cl( i -Int( j -Cl(A))) j -Cl(A). Therefore, j -Cl(A) = j -Cl( i Int( j -Cl (A))). The converse is obvious. De nition 3.11. A bitopological space (X; 1 ; 2 ) is said to be pairwise extremally disconnected [1] if j -Cl(A) 2 i for every A 2 i . Proposition 3.12. Let (X; 1 ; 2 ; I ) be a pairwise extremally disconnected space. Then if A is (i; j )- -I -open, then A is (i; j )-preopen in Proof.
X.
Proof. Let A be (i; j )- -I -open set of X , we have A j -Cl( i -Int( j Cl (A))). Since X is pairwise extremally disconnected, for i -Int( j Cl (A)) 2 , we have -Cl( -Int( -Cl (A))) 2 . So, we have A i
j
i
j
i
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j -Cl( i -Int( j -Cl (A))) i -Int( j -Cl( i -Int( j -Cl (A)))) i -Int( j Cl( j -Cl (A))) i -Int( j -Cl(A [ A )) = i -Int( j -Cl(A) [ j -Cl(A )) i -Int( j -Cl(A)); hence A is (i; j )-preopen in X . An ideal bitopological space is said to satisfy the condition (A) if U \ j Cl (A) j -Cl (U \ A) for every U 2 i . Theorem 3.13. Let (X; 1 ; 2 ; I ) be a pairwise extremally disconnected space which satis es the condition A. If A is (i; j )-semi-I -open and B is (i; j )-pre-I -open, then A \ B is (i; j )- -I -open. Proof. Let A be (i; j )-semi-I -open and B an (i; j )-pre-I -open set of X . Then A \ B j -Cl ( i -Int(A)) \ i -Int( j -Cl (B )) j -Cl ( i Int(A) \ i -Int( j -Cl (B )) = j -Cl ( i -Int( i -Int(A)) \ j -Cl (B ))) j Cl ( i -Int( j -Cl ( i -Int(A) \ B ))) j -Cl ( i -Int( j -Cl (A \ B ))) j Cl( i -Int( j -Cl (A \ B ))). Thus, A \ B is (i; j )- -I -open in X . De nition 3.14. In an ideal bitopological space (X; 1 ; 2 ; I ), A X is said to be (i; j )- -I -closed if X nA is (i; j )- -I -open in X , i; j = 1; 2 and i 6= j . Theorem 3.15. If A is an (i; j )- -I -closed set in an ideal bitopological space (X; 1 ; 2 ; I ) if and only if j -Int( i -Cl( j -Int (A))) A.
Theorem 3.16. A subset A of an ideal bitopological space (X; 1 ; 2 ; I ) is (i; j )- -I -closed, then j -Int( i -Cl ( j -Int(A))) A Proof. The proof follows from the fact that Cl (A) Cl(A) for every subset A of X . Theorem 3.17. Arbitrary intersection of (i; j )- -I -closed sets is always (i; j )- -I -closed. Proof. Follows from Theorems 3.9 and 3.16. De nition 3.18. Let (X; 1 ; 2 ; I ) be an ideal bitopological space, S a Proof.
The proof follows from the de nitions.
subset of X and x be a point of X . Then (i) x is called an (i; j )- -I -interior point of S if there exists V 2 (i; j )- I O(X; 1 ; 2 ) such that x 2 V S . ii) the set of all (i; j )- -I -interior points of S is called (i; j )- -I interior of S and is denoted by (i; j )- I Int(S ).
Theorem 3.19. Let A and B be subsets of (X; 1 ; 2 ; I ). Then the following properties hold: (i) (i; j )- I Int(A) = [fT : T A and A 2 (i; j )- I O(X )g. (ii) (i; j )- I Int(A) is the largest (i; j )- -I -open subset of X contained in A. (iii) A is (i; j )- -I -open if and only if A = (i; j )- I Int(A). (iv) (i; j )- I Int((i; j )- I Int(A)) = (i; j )- I Int(A).
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(v) If A B , then (i; j )- I Int(A) (i; j )- I Int(B ). (vi) (i; j )- I Int(A) [ (i; j )- I Int(B ) (i; j )- I Int(A [ B ). (vii) (i; j )- I Int(A \ B ) (i; j )- I Int(A) \ (i; j )- I Int(B ). Proof. (i). Let x 2 [fT : T A and A 2 (i; j )- I O(X )g. Then, there exists T 2 (i; j )- I O(X; x) such that x 2 T A and hence x 2 (i; j ) I Int(A). This shows that [fT : T A and A 2 (i; j )- I O(X )g (i; j )- I Int(A). For the reverse inclusion, let x 2 (i; j )- I Int(A). Then there exists T 2 (i; j )- I O(X; x) such that x 2 T A. we obtain x 2 [fT : T A and A 2 (i; j )- I O(X )g. This shows that (i; j )- I Int(A) [fT : T A and A 2 (i; j )- I O(X )g. Therefore, we obtain (i; j )- I Int(A) = [fT : T A and A 2 (i; j )- I O(X )g. The proof of (ii) (v) are obvious. (vi). Clearly, (i; j )- I Int(A) (i; j )- I Int(A[B ) and (i; j )- I Int(B ) (i; j )- I Int(A [ B ). Then by (v) we obtain (i; j )- I Int(A) [ (i; j ) I Int(B ) (i; j )- I Int(A [ B ). (vii). Since A \ B A and A \ B B , by (v), we have (i; j )- I Int(A \ B ) (i; j )- I Int(A) and (i; j )- I Int(A \ B ) (i; j )- I Int(B ). By (v) (i; j )- I Int(A \ B ) (i; j )- I Int(A) \ (i; j )- I Int(B ). De nition 3.20. Let (X; 1 ; 2 ; I ) be an ideal bitopological space, S a
subset of X and x be a point of X . Then (i) x is called an (i; j )- -I -cluster point of S if V \ S 6= ; for every V 2 (i; j )- I O(X; x). (ii) the set of all (i; j )- -I -cluster points of S is called (i; j )- -I closure of S and is denoted by (i; j )- I Cl(S ).
Theorem 3.21. Let A and B be subsets of (X; 1 ; 2 ; I ). Then the following properties hold: (i) (i; j )- I Cl(A) = \fF : A F and F 2 (i; j )- I C (X )g. (ii) (i; j )- I Cl(A) is the smallest (i; j )- -I -closed subset of X containing A. (iii) A is (i; j )- -I -closed if and only if A = (i; j )- I Cl(A). (iv) (i; j )- I Cl((i; j )- I Cl(A) = (i; j )- I Cl(A). (v) If A B , then (i; j )- I Cl(A) (i; j )- I Cl(B ). (vi) (i; j )- I Cl(A [ B ) = (i; j )- I Cl(A) [ (i; j )- I Cl(B ). (vii) (i; j )- I Cl(A \ B ) (i; j )- I Cl(A) \ (i; j )- I Cl(B ).
Suppose that x 2= (i; j )- I Cl(A). Then there exists F 2 (i; j )- I O(X ) such that V \ S 6= ;. Since X nV is (i; j )- -I -closed set containing A and x 2= X nV , we obtain x 2= \fF : A F and F 2 (i; j )- I C (X )g. Then there exists F 2 (i; j )- I C (X ) such that A F and x 2= F . Since X nV is (i; j )- -I -closed set containing x, we obtain (X nF ) \ A = ;. This shows that x 2= (i; j )- I Cl(A). Therefore, we obtain (i; j )- I Cl(A) = \fF : A F and F 2 (i; j )- I C (X ). The other proofs are obvious. Proof. (i).
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Theorem 3.22. Let (X; 1 ; 2 ; I ) be an ideal bitopological space and A X . A point x 2 (i; j )- I Cl(A) if and only if U \ A 6= ; for every U 2 (i; j )- I O(X; x). Proof. Suppose that x 2 (i; j )- I Cl(A). We shall show that U \ A 6= ; for every U 2 (i; j )- I O(X; x). Suppose that there exists U 2 (i; j ) I O(X; x) such that U \ A = ;. Then A X nU and X nU is (i; j )- I -closed. Since A X nU , (i; j )- I Cl(A) (i; j )- I Cl(X nU ). Since x 2 (i; j )- I Cl(A), we have x 2 (i; j )- I Cl(X nU ). Since X nU is (i; j )- -I -closed, we have x 2 X nU ; hence x 2= U , which is a contradiction that x 2 U . Therefore, U \ A 6= ;. Conversely, suppose that U \ A 6= ; for every U 2 (i; j )- I O(X; x). We shall show that x 2 (i; j )- I Cl(A). Suppose that x 2= (i; j )- I Cl(A). Then there exists U 2 (i; j )- I O(X; x) such that U \ A = ;. This is a contradiction to U \ A 6= ;; hence x 2 (i; j )- I Cl(A). Theorem 3.23. Let (X; 1 ; 2 ; I ) be an ideal bitopological space and A X . Then the following propeties hold: (i) (i; j )- I Int(X nA) = X n(i; j )- I Cl(A); (i) (i; j )- I Cl(X nA) = X n(i; j )- I Int(A). Proof. (i). Let x 2 (i; j )- I Cl(A). There exists V 2 (i; j )- I O(X; x) such that V \ A 6= ;; hence we obtain x 2 (i; j )- I Int(X nA). This shows that X n(i; j )- I Cl(A) (i; j )- I Int(X nA). Let x 2 (i; j ) I Int(X nA). Since (i; j )- I Int(X nA) \ A = ;, we obtain x 2= (i; j ) I Cl(A); hence x 2 X n(i; j )- I Cl(A). Therefore, we obtain (i; j ) I Int(X nA) = X n(i; j )- I Cl(A).
(ii). Follows from (i).
Proposition 3.24. The product of two (i; j )- -I -open sets is (i; j )- -
I -open. Proof.
The proof follows from Lemma 3.3 of [12].
4. (i; j )- -I -continuous functions De nition 4.1. A function f : (X; 1 ; 2 ; I ) ! (Y; 1 ; 2 ) is said to be (i; j )- -I -continuous if the inverse image of every i -open set of Y is (i; j )- -I -open in X , where i 6= j , i; j =1, 2. Proposition 4.2. (i) Every (i; j )-b-I -continuous function is (i; j ) -I -continuous but not conversely.
(ii) Every (i; j )- -I -continuous function is (i; j )-semi-precontinuous but not conversely.
The proof follows from Proposition 3.3 and Example 3.4. Corollary 4.3. (i) Every (i; j )- -I -continuous function is (i; j )Proof.
-I -continuous but not conversely.
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(ii) Every (i; j )-semi-I -continuous function is (i; j )- -continuous but not conversely.
(iii) Every (i; j )-pre-I -continuous function is (i; j )- -I -continuous but not conversely.
Theorem 4.4. For a function f : (X; 1 ; 2 ; I ) ! (Y; 1 ; 2 ), the following statements are equivalent: (i) f is pairwise -I -continuous; (ii) For each point x in X and each i -open set F in Y such that f (x) 2 F , there is a (i; j )- -I -open set A in X such that x 2 A, f (A) F ; (iii) The inverse image of each i -closed set in Y is (i; j )- -I -closed in X ; (iv) For each subset A of X , f ((i; j )- I Cl(A)) i -Cl(f (A)); (v) For each subset B of Y , (i; j )- I Cl(f 1 (B )) f 1 (i -Cl(B )). Proof. (i) ) (ii):
Let x 2 X and F be a j -open set of Y containing f (x). By (i), f 1 (F ) is (i; j )- -I -open in X . Let A = f 1 (F ). Then x 2 A and f (A) F . (ii) ) (i): Let F be j -open in Y and let x 2 f 1 (F ). Then f (x) 2 F . By (ii), there is an (i; j )- -I -open set Ux in X such that x 2 Ux and f (Ux ) F . Then x 2 Ux f 1 (F ). Hence f 1 (F ) is (i; j )- -I -open in X . (i) , (iii): This follows due to the fact that for any subset B of Y , f 1 (Y nB ) = X nf 1 (B ). (iii) ) (iv): Let A be a subset of X . Since A f 1 (f (A)) we have A f 1 (j -Cl(f (A))). Now, (i; j )- I Cl(f (A)) is j -closed in Y and hence f 1 (j -Cl(A)) f 1 (j -Cl(f (A))), for (i; j )- I Cl(A) is the smallest (i; j )- -I -closed set containing A. Then f ((i; j )- I Cl(A)) j -Cl(f (A)). (iv) ) (iii): Let F be any (i; j )- -I -closed subset of Y . Then f ((i; j ) I Cl(f 1 (F ))) (i; j )-i -Cl(f (f 1 (F ))) = (i; j )-i -Cl(F ) = F . Therefore, (i; j )- I Cl(f 1 (F )) f 1 (F ). Consequently, f 1 (F ) is (i; j )- I -closed in X . (iv) ) (v): Let B be any subset of Y . Now, f ((i; j )- I Cl(f 1 (B ))) (i; j )-i -Cl(f (f 1 (B ))) i -Cl(B ). Consequently, (i; j )- I Cl(f 1 (B )) f 1(i-Cl(B )). (v) ) (iv): Let B = f (A) where A is a subset of X . Then, (i; j ) I Cl(A) (i; j )- I Cl(f 1 (B )) f 1 (i -Cl(B )) = f 1 (i -Cl(f (A))). This shows that f ((i; j )- I Cl(A)) i -Cl(f (A)). If I = f;g in Theorem 4.4, we get the following Corollary 4.5 ([8], Theorem 5.1). For a function f : (X; 1 ; 2 ) ! (Y; 1 ; 2 ), the following statements are equivalent: (i) f is pairwise semi-precontinuous;
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(ii) For each point x in X and each i -open set F in Y such that f (x) 2 F , there is a (i; j )-semi-preopen set A in X such that x 2 A, f (A) F ; (iii) The inverse image of each i -closed set in Y is (i; j )-semi-
preclosed in X ; (iv) For each subset A of X , f ((i; j )-sp Cl(A)) i -Cl(f (A)); (v) For each subset B of Y , (i; j )-sp Cl(f 1 (B )) f 1 (i -Cl(B )). Theorem 4.6. Let f : (X; 1 ; 2 ; I ) ! (Y; 1 ; 2 ) be a function. If g : (X; 1 ; 2 ; I ) ! (X Y; 1 2 ) de ned by g(x) = (x; f (x)) is a pairwise -I -continuous function, then f is pairwise -I -continuous. Proof. Let V be a i -open set of Y . Then f 1 (V ) = X \ f 1 (V ) =
g 1 (X V ). Since g is a pairwise -I -continuous function and X V is a i i -open set of X Y , f 1 (V ) is a (i; j )- -I -open set of X . Hence f is pairwise -I -continuous. De nition 4.7. A subset K of an ideal bitopological space (X; 1 ; 2 ; I ) is said to be pairwise -I -compact relative to X , if for every cover fU : 2 g of K by (i; j )-S -I -open sets of X , there exists a nite subset 0 of such that K n fU : 2 0 g 2 I . The space (X; 1 ; 2 ; I ) is said to be pairwise -I -compact if X is pairwise -I -compact subsets of X . De nition 4.8. A subset K of an ideal bitopological space (X; 1 ; 2 ; I ) is said to be pairwise countable -I -compact relative to X , if for every cover fU : 2 g of K by countable (i; j )-S -I -open sets of X , there exists a nite subset 0 of such that K n fU : 2 0 g 2 I . The space (X; 1 ; 2 ; I ) is said to be pairwise countable -I -compact if X is pairwise countable -I -compact subset of X . De nition 4.9. A subset K of an ideal bitopological space (X; 1 ; 2 ; I ) is said to be pairwise -I -Lindelof relative to X , if for every cover fU : 2 g of K by (i; j )-S -I -open sets of X , there exists a nite subset 0 of such that K n fU : 2 0 g 2 I . The space (X; 1 ; 2 ; I ) is said to be pairwise -I -Lindelof if X is pairwise -I -Lindelof subset of X. Theorem 4.10. If f : (X; 1 ; 2 ; I ) ! (Y; 1 ; 2 ) is a pairwise -I continuous surjection and (X; 1 ; 2 ; I ) is pairwise -I -compact,then (Y; 1 ; 2 ; f (I )) is pairwise f (I )-compact. Proof. Let fV : 2 g be a i -open cover of Y . Then ff 1 (V ) : 2 g is an (i; j )- -I -openScover of X and hence, there exist a nite subset 0 of such that X n ff 1 (VS ) : 2 0 g 2 I . Since f is surjective, S Y n fV : 2 0 g = f (X n ff 1 (V ) : 2 0 g) 2 I . Therefore, (Y; 1 ; 2 ; f (I )) is pairwise f (I )-compact. Theorem 4.11. If f : (X; 1 ; 2 ; I ) ! (Y; 1 ; 2 ) is a pairwise - continuous surjection and (X; 1 ; 2 ; I ) is pairwise -I -Lindelof, then (Y; 1 ; 2 ; f (I )) is pairwise f (I )-Lindelof.
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The proof is similar to previous theorem. Theorem 4.12. If f : (X; 1 ; 2 ; I ) ! (Y; 1 ; 2 ) is a pairwise I -continuous surjection and (X; 1; 2; I ) is pairwise countable -I compact, then (Y; 1 ; 2 ; f (I )) is countably f (I )-compact. Proof. The proof is similar to previous theorem. De nition 4.13. A bitopological space (X; 1 ; 2 ) is said to be pairwise connected [10] if it cannot be expressed as the union of two nonempty Proof.
disjoint sets U and V such that U is i -open and V is j -open, where i; j = f1; 2g.
De nition 4.14. An ideal bitopological space (X; 1 ; 2 ; I ) is said to be pairwise -I -connected if it cannot be expressed as the union of two nonempty disjoint sets U and V such that U is (i; j )- -I -open and V is (i; j )- -I -open. Theorem 4.15. Let f : (X; 1 ; 2 ; I ) ! (Y; 1 ; 2 ) is pairwise -I continuous surjection and (X; 1 ; 2 ; I ) is pairwise -I -connected, then (Y; 1 ; 2 ) is pairwise connected. Proof. Suppose Y is not pairwise connected, Then Y = A [ B where A \ B = ;, A 6= ;, B 6= ; and A 2 i , B 2 j . Since f is pairwise -I -continuous f 1 (A) 2 (i; j )- I O(X ) and f 1 (B ) 2 (i; j )- I O(X ), such that f 1 (A) 6= ;, f 1 (B ) 6= ;. f 1 (A) \ f 1 (B ) = ; and f 1 (A) [ f 1 (B ) = X , which implies that X is not pairwise -I -connected. References
[1] G. Balasubramanian, Extremally disconnected bitopological spaces, Bull. Cal. Math. Soc., 83, 1991, 247-252. [2] M. Caldas, S. Jafari and N. Rajesh, Preopen sets in ideal bitopological spaces (submited). [3] M. Caldas, S. Jafari and N. Rajesh, Semiopen sets in ideal bitopological spaces (submitted). [4] M. Caldas, S. Jafari and N. Rajesh, b-open sets in ideal bitopological spaces (under preparation). [5] M. Caldas, S. Jafari and N. Rajesh, Properties of ideal bitopological -open sets (submitted). [6] D. Jankovic and T. R. Hamlett, New topologies from old via ideals, American Math. Monthly, 97(1990), 295-310. [7] M. Jelic, Feeble P -continuious mappings, Rend. Circ Math. Palermo, 24(1990), 387-. [8] F. Khedr, S. Al-Aree and T. Noiri, Precontinuity and semi-precontinuity in bitopological spaces, Indian J. Pure Appl. Math. 23(9) (1992), 624-633. [9] K. Kuratowski, Topology, Academic press, New York, (1966). [10] W. J. Pervine, Connectedness in bitopological spaces, Ind. Math., 29(1967), 369-. [11] R. Vaidyanathaswamy, The localisation theory in set topology, Proc. Indian Acad. Sci., 20(1945), 51-61. [12] S. Yuksel, A. H. kocaman and A. Acikgoz, On -I -irresolute functions, Far East J. Math. Sci., 26(3)(2007), 673-684
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M. CALDAS, S. JAFARI AND N. RAJESH
Departamento de Matematica Aplicada, Universidade Federal Fluminense,, Rua Mario Santos Braga, S/n, 24020-140, Niteroi, RJ Brasil
E-mail address :
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Department of Mathematics, College of Vestsjaelland South, Herrestraede, 11, 4200 Slagelse, Denmark
E-mail address :
[email protected]
Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur613005, Tamilnadu, India.
E-mail address : nrajesh
[email protected] 
