Molodtsov's Soft Topological Structures

Abstract

This study explores foundational concepts of soft topological structures introduced by Molodtsov in 2015. These include the notions of proximity mappings, $\tilde{\tau}$-neighborhood of an element, $\tilde{\tau}$-neighborhood of a set, $\tilde{\tau}$-closure, $\tilde{\tau}$-interior, $\tilde{\tau}$-boundary, $\tilde{\tau}$-border set, $\tilde{\tau}$-hidden set, $\tilde{\tau}$-accumulation points, $\tilde{\tau}$-cluster points, $\tilde{\tau}$-isolated points, and $\tilde{\tau}$-separated points. In addition to providing a theoretical basis for these notions, the study defines $\tilde{\tau}$-disjointness, $\tilde{\tau}$-exterior, and $\tilde{\tau}$-condensation points of a set. It further proposes $\tilde{\tau}$-$C$-closed, $\tilde{\tau}$-$B$-open, $\tilde{\tau}$-$T$-closed, and $\tilde{\tau}$-$I$-open sets. Based on these, this paper constructs the corresponding $\tilde{\tau}$-$C$-, $\tilde{\tau}$-$B$-, $\tilde{\tau}$-$T$-, and $\tilde{\tau}$-$I$-soft topologies. Additionally, the present study investigates the fundamental properties of these newly proposed concepts. Furthermore, it clarifies all concepts with detailed examples. Finally, the paper outlines potential directions for further research on these aspects.