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

• DOI: 10.1007/s40314-020-01161-3
• Publication Date: 2020-05-09T04:02:48Z
• Authors: M. E. El-Shafei; T. M. Al-shami
• Keywords: Separation axiom; Soft set; Equivalence relation

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 $$T_i$$ -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 $$T_i$$ -spaces, p-soft $$T_i$$ -spaces and e-soft $$T_i$$ -spaces. We illustrate the relationships between w-soft $$T_i$$ -spaces with the help of examples and discuss some sufficient conditions of soft topological spaces to be w-soft $$T_i$$ -spaces. Additionally, we point out that stable or soft regular spaces are sufficient conditions for the equivalence among the concepts of soft $$T_i$$ , p-soft $$T_i$$ and w-soft $$T_i$$ . We highlight on explaining the links between w-soft $$T_i$$ -spaces and their parametric topological spaces and studying the role of enriched spaces in these links. Furthermore, we prove that w-soft $$T_i$$ -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.