A New Operator via Regular Open Sets in a New Topological Structure

In this paper, we will use the family of regular open sets in a topological space (Z, ? ) to define an operator ?R : 2Z ? 2Z by ?R(F) = {s ? Z : ? D ? RO(Z, s) with (D ? F )c /? P} in frame of primal topological spaces. Then we introduce the notion of topology ?-compatible for a primal in a primal topological space and study some of its properties. Finally, we use the concept of ?-semi-open sets to provide additional properties for the operators (? R) and ?R(F ), and we add many illustrative examples that help clarify the relationships between the concepts that are presented.