Topologically indistinguishable relations and separation axioms - Sibel Demiralp; Tareq M. Al-shami; Fuad A. Abushaheen; Alaa M. Abd El-latif (American Institute of Mathematical Sciences (AIMS), 2024)
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.
Keywords: Rough Sets Theory and Applications; Social Sciences; Geometry; Separation axiom; Management Science and Operations Research; Decision Sciences; Automata Theory; Fuzzy Logic and Residuated Lattices; Topological Spaces; QA1-939; FOS: Mathematics; Axiom; equivalence relation; Probabilistic Rough Sets; Mathematical economics; Statistics; Application of Soft Set Theory in Decision Making; Association Rules Mining; Probability Theory; Computer science; Computational Theory and Mathematics; Combinatorics; separation axioms; Computer Science; Physical Sciences; Separation (statistics); $ f\mathcal{t} $-topological distinguishability; Mathematics; 10. No inequality; 4. Education; 15. Life on land; 16. Peace & justice