Applications of partial belong and total non-belong relations on soft separation axioms and decision-making problem

This study introduces a new family of soft separation axioms and a real-life application utilizing partial belong and natural non-belong relations. First, we initiate the concepts of w-soft Ti-spaces (i=0,1,2,3,4) with respect to distinct ordinary points. These concepts generate a wider family of soft spaces compared with soft Ti-spaces, p-soft Ti-spaces and e-soft Ti-spaces. We illustrate the relationships between w-soft Ti-spaces with the help of examples and discuss some sufficient conditions of soft topological spaces to be w-soft Ti-spaces. Additionally, we point out that stable or soft regular spaces are sufficient conditions for the equivalence among the concepts of soft Ti, p-soft Ti\ and w-soft Ti.We highlight on explaining the links between w-soft Ti-spaces and their parametric topological spaces and studying the role of enriched spaces in these links. Furthermore, we prove that w-soft Ti-spaces are hereditary and topological properties, and they are preserved under finite product soft spaces. Finally, we propose an algorithm to bring out the optimal choices. This algorithm is based on dividing the whole parameters set into parameter sets and then apply a partial belong relation in the favorite soft sets. This application is supported with an interesting example to show how to implement this algorithm.