Topologically indistinguishable relations and separation axioms

This study focuses on defining separation axioms for sets without an inherent topological structure. By utilizing a mapping to relate such sets to a topological space, we first define a distinguishable relation over the universal set with respect to the neighborhood systems inspired by a topology of the co-domain set and elucidate its basic properties. To facilitate the way of discovering this distinguishable relation, we initiate a color technique for the equivalence classes inspired by a given topology. Also, we provide an algorithm to determine distinguishable members (or objects) under study. Then, we establish a framework for introducing separation properties within these structureless sets and examine their master characterizations. To better understand the obtained results and relationships, we display some illustrative instances.