Applications of a Conditional Preopen Sets in Bitopological Spaces: Topological Space in Mathematics

The purpose of the present work is to introduce and investigate a new class of sets called (i, j)-Ps? open sets, and use this class to define and study new concepts in bitopological spaces such as continuity and separation axioms. At the beginning of this work, we define the class of (i, j)-Ps? open sets which contained in the class of j-preopen sets and also contained in the class of (i, j)-gp? open sets. It is shown that the family of (i, j)-Ps? open sets form a supratopology on X. We prove that the family of (i, j)-Ps? open sets and the family of j-preopen sets are identical when (X, ?i) are semi-T1-spaces. Finally, some separation axioms such as T0, T1 and T2 spaces are defined in bitopological spaces, also R0, R1 and Urysohn spaces are defined and the relation between them are found by using the new type of graph functions called (i, j)-Ps ? closed graph. It is noticed that if (X, ?1, ?2) is (i, j)-Ps?Tk, then it is (i, j)-Ps?Tk-1, for k=1, 2. It is proved that a bitopological space (X, ?1, ?2) is (i, j)-Ps?T1 if the (i, j)-Ps? derived set of every point of X is empty.