Sets Related to Openness and Continuity Decompositions in Primal Topological Spaces

This paper introduces and investigates several new classes of sets called P-α-open sets, Psemiopen sets, P-preopen sets, and P-β-open sets within the framework of primal topological spaces. Their properties and relationships with other open set generalizations are studied through examples. Additionally, the concepts of PR-sets and PRα -sets are defined and their characteristics examined. Also, the notions of P-α-continuous, P-semicontinuous, P-precontinuous and P-β-continuous mappings are initiated and their features and main characterizations determined. A new class of sets called ΨP -sets is also introduced in primal topological spaces using the ΨP -operator. Their properties and relationships between Ψ_ P -sets, α-open, semi-open, and pre-open are investigated. Theorems on arbitrary unions and finite intersections of ΨP are discussed.