Abstract:

The purpose of the present work is to introduce and investigate a new class of sets called (i, j)-P_{s}– open sets, and use this class to define and study new classes of concepts in bitopology such as continuity and separation axioms.

At the beginning of this work, are define the class of (i, j)-P_{s}– open sets which is contained in the class of j-preopen and also contained in (i, j)-gp– open sets. It is shown that the family of (i, j)-P_{s}– open sets forms a supratopology on X. we prove that the family of (i, j)-P_{s}– open sets and j-preopen sets are identical where (X, τ_{i}) is semi-T_{1}-spaces. Also noticed that the union of family of (i, j)-P_{s}– open sets are also (i, j)-P_{s}– open sets. We prove that the family of (i, j)-P_{s}– open sets imply the family of i-open sets where (X, τ_{i}) is locally indiscrete. It is shown that if the family of j-preopen sets in a bitopological space X forms a topology, then the family of (i, j)-P_{s}– open sets is also topology on X. It is also we noticed that if AYX and A is (i, j)-P_{s}– open set of X, then (i, j)-P_{s}Cl_{y}(A)(i, j)-P_{s}Cl(A), also if AYX and Y is i-regular open and j-regular open, then (i, j)-P_{s}Cl(A)Y=(i, j)-P_{s}Cl_{y}(A), and for any subset A of X we have (i, j)-P_{s}D((i, j)-P_{s}D(A))\A(i, j)-P_{s}D(A). All results above are in chapter two.

In chapter three, we introduce some strong types of precontinuity and almost precontinuity in bitopological spaces. Moreover a type of contra precontinuity is defined. Several properties and characterizations of these functions are obtained in terms of various types of sets. If a function :(X, τ_{1}, τ_{2})(Y,σ_{1}, σ_{2}) is (i, j)-P_{s} – continuous then, it is almost (i, j)-P_{s} – continuous, but the converse is true Y is i-semi-regular. The concepts of contra (i, j)-P_{s} – continuous and j-contra precontinuity are equivalent for a function :(X, τ_{1}, τ_{2})(Y,σ_{1}, σ_{2}) if (X, τ_{i}) is semi-T_{1}-spaces. If a function :(X, τ_{1}, τ_{2})(Y,σ_{1}, σ_{2}) is contra (i, j)-P_{s} – continuous, then it is j-contra continuous when (X, τ_{j}) is door spaces.

Finally, in chapter four, we define some separation axioms like T_{0}, T_{1} and T_{2} spaces in bitopological spaces, we also defined R_{0}, R_{1} and Urysohn spaces and we find the relation between them by using the new type of graph function called (i, j)-P_{s} – closed graph. We noticed that if (X, τ_{1}, τ_{2}) is (i, j)-P_{s}–T_{i}, then it is (i, j)-P_{s}–T_{i-1}, for i=1, 2. Also we shows that every subspace of an (i, j)-P_{s}–T_{1}– space is also (i, j)-P_{s}–T_{1}. We prove that a bitopological space (X, τ_{1}, τ_{2}) is (i, j)-P_{s}–T_{1} if the (i, j)-P_{s}– derived set of a point b is empty, for all b in X. We shows that every j-α-open subspace of (i, j)-P_{s}–Urysohn is also (i, j)-P_{s}–Urysohn, where (X, τ_{i}) is semi-T_{1}-spaces. Also we prove that a bitopological space (X, τ_{1}, τ_{2}) is (i, j)-P_{s}–R_{0} if and only if the (i, j)-P_{s}– closure of x is coincides with (i, j)-P_{s}– kernel of x, for all x in X.