Two New Families of Supra-Soft Topological Spaces Defined by Separation Axioms

This paper contributes to the field of supra-soft topology. We introduce and investigate supra p p -soft T j and supra p t -soft T j -spaces ( j = 0 , 1 , 2 , 3 , 4 ) . These are defined in terms of different ordinary points; they rely on partial belong and partial non-belong relations in the first type, and partial belong and total non-belong relations in the second type. With the assistance of examples, we reveal the relationships among them as well as their relationships with classes of supra-soft topological spaces such as supra t p -soft T j and supra t t -soft T j -spaces ( j = 0 , 1 , 2 , 3 , 4 ) . This work also investigates both the connections among these spaces and their relationships with the supra topological spaces that they induce. Some connections are shown with the aid of examples. In this regard, we prove that for i = 0 , 1 , possessing the T i property by a parametric supra-topological space implies possessing the p p -soft T i property by its supra-soft topological space. This relationship is invalid for the other types of soft spaces introduced in previous literature. We derive some results of p p -soft T i -spaces from the cardinality numbers of the universal set and a set of parameters. We also demonstrate how these spaces behave as compared to their counterparts studied in soft topology and its generalizations (such as infra-soft topologies and weak soft topologies). Moreover, we investigated whether subspaces, finite product spaces, and soft S