On Soft Separation Axioms and Their Applications on Decision-Making Problem

Abstract

In this work, we introduce new types of soft separation axioms called $pt$ -soft $\alpha$ regular and $pt$ -soft $\alpha T_i$ -spaces $\left(i=\mathrm{0,1,2,3,4}\right)$ using partial belong and total nonbelong relations between ordinary points and soft $\alpha$ -open sets. These soft separation axioms enable us to initiate new families of soft spaces and then obtain new interesting properties. We provide several examples to elucidate the relationships between them as well as their relationships with $e$ -soft $T_i$ , soft $\alpha T_i$ , and $tt$ -soft $\alpha T_i$ -spaces. Also, we determine the conditions under which they are equivalent and link them with their counterparts on topological spaces. Furthermore, we prove that $pt$ -soft $\alpha T_i$ -spaces $\left(i=\mathrm{0,1,2,3,4}\right)$ are additive and topological properties and demonstrate that $pt$ -soft $\alpha T_i$ -spaces $\left(i=\mathrm{0,1,2}\right)$ are preserved under finite product of soft spaces. Finally, we discuss an application of optimal choices using the idea of $pt$ -soft $T_i$ -spaces $\left(i=\mathrm{0,1,2}\right)$ on the content of soft weak structure. We provide an algorithm of this application with an example showing how this algorithm is carried out. In fact, this study represents the first investigation of real applications of soft separation axioms.