On bipreordered approximation spaces

Abstract

We used preordered relations to define a bipreordered space and hence bitopological space and introduced a condition (*) on these relations such that R(A ∪ B) = R(A) ∪ R(B), where R(A) = R-1 (A) ∩ R-2 (A), and hence we get a topology τR12 on X satisfies A = R (A) = R-1 (A) ∩ R-2 (A) = {x ∈ X: xR1 ∩ xR2 ∩ A ≠ φ} = A-1 ∩ A-2 and τR12 = τR1∩R2 = τR1 VτR2.We deal with bitopological spaces (X, τ1, τ2) which satisfying a certain condition (**) and proved that the family of all such bitopological spaces BTS** is equivalent to the family of all bipreordered spaces BPS*.