On weakly b-continuous functions in bitopological spaces

In this article we introduce the notion of weakly b-continuous functions in bitopological spaces as a generalization of b-continuous functions. We prove several properties of these functions. AMS Classification No : 54A10; 54C10; 54C08; 54D15.

In this paper, we introduce the notion of weakly b-continuous functions in bitopological spaces and investigate their different properties.

Preliminaries
Throughout the present paper (X,) denotes a topological space and (X, 1 , 2 ) denotes a bitopological space on which no separation axioms are assumed. Let (X, 1 , 2 ) be a bitopological space and A be a subset of X. The closure(resp. interior) of A with respect to the topology  i (i =1, 2) will be denoted by i Cl(A) (resp. i Int(A)). Now we list some known definitions and results those will be used throughout this article.
(ii) the arbitrary intersection of (i, j)-b-closed sets is (i, j)-b-closed.
Lemma 2.2. Let (X, 1 , 2 ) be a bitopological space and A be a subset of X. ( Lemma 2.3. Let (X, 1 , 2 ) be a bitopological space and A be a subset of X. Then Definition 4. Let (X, 1 , 2 ) be a bitopological space and A be a subset of X. A point x of X is said to be in the (i, j)--closure of A, denoted by (i, j)- Lemma 2.5. For a subset A of a bitopological space (X, 1 , 2 ), the following properties hold: (2) (i, j)-weakly continuous if for each xX and Now we introduce the following definition in this article.
Definition 6. A function f :
(4)(1) Let V be any  j -open subset of Y. Then V j Int (i Cl(V )) = j Int((i, j)-Cl  (V )), by Lemma 2.6. Now by hypothesis we have Theorem 6. Let f : (X, 1 , 2 )  (Y, 1 , 2 ) be a function. If f is (i, j)-weakly b-continuous then inverse image of every (i, j)--closed set of Y is (i, j)b-closed in X, for all i, j =1, 2.

Some further properties
Definition 2. Let (X, 1 , 2 ) be a bitopological space and A be a subset of X. Then (i, j)-b-frontier of A is defined as follows: . Theorem 2. The set of all points x of X at which a function f : (X, 1 , 2 )  (Y, 1 , 2 ) is not (i, j)-weakly b-continuous is identical with the union of the (i, j)-b-frontiers of the inverse images of the  j -closure of  i -open set of Y containing f(x), for all i, j=1, 2.
Proof: Let x be a point of X at which f(x) is not (i, j)-weakly b-continuous. Then there exists a  iopen subset B of Y containing f(x) such that A X -( f -1 ( j Cl(B))) for every (i, j)-b-open subset A of X containing x. By Lemma 2.4, we have x(i, j)-b ClX -( f -1 ( j Cl(B))). Since xf -1 ( j Cl(B)), we have x(i, j)-b Cl(f -1 ( j Cl(B))). Therefore we have x(i, j)-b Cl(f -1 ( j Cl(B))) (i, j)-b ClX -( f -1 ( j Cl(B))). Hence x(i, j)-b Fr( f -1 ( j Cl(B))).
Conversely if f is (i, j)-weakly b-continuous at x, then for each  i -open set B of Y containing f(x), there exists an (i, j)-b-open subset A containing x such that f(A)  j Cl(B) and hence xA  f -1 ( j Cl(B)). Therefore x(i, j)-b Int( f -1 ( j Cl(B))). Which is a contradiction to x(i, j)-b Fr(f -1 ( j Cl(B))).
Definition 3. A bitopological space (X, 1 , 2 ) is said to be pairwise b-T 2 if for each pair of distinct introduced. The notion of b-frontier of a subset in a bitopological space has been introduced. It is shown, if (Y, 1 , 2 ) is a pairwise Urysohn and f : (X, 1 , 2 )  (Y, 1 , 2 ) is (i, j)-weakly b-continuous injection, then (X, 1 , 2 ) is pairwise b-T 2 . These notions can be applied for investigating many other properties and some properties relative to separation axioms.