Generalized open sets and closure operators via point-to-neighborhood assignments

Abstract

We equip a topological space (Formula presented.) with a function (Formula presented.) satisfying the single axiom (Formula presented.). The resulting triple (Formula presented.), which we call an aura topological space, provides a point-to-open-set assignment that differs from all existing auxiliary structures in topology—ideals, filters, grills, primals, and the various non-classical frameworks based on fuzzy, soft, or neutrosophic sets. The aura-closure operator (Formula presented.) is shown to be an additive Čech closure operator; it satisfies extensivity, monotonicity, and finite additivity, but idempotency fails in general. Iterating (Formula presented.) transfinitely yields a Kuratowski closure whose topology (Formula presented.) satisfies (Formula presented.), where (Formula presented.) is the collection of all (Formula presented.) -open sets. We introduce (Formula presented.) -semi-open, (Formula presented.) -pre-open, (Formula presented.) - (Formula presented.) -open, and (Formula presented.) - (Formula presented.) -open sets, determine the complete hierarchy among these classes and their classical counterparts, and separate all non-coinciding classes by counterexamples on finite spaces as well as on the real line. The notions of (Formula presented.) -convergence of sequences and the corresponding continuity notions and their decompositions are studied. Separation axioms (Formula presented.) - (Formula presented.) (Formula presented.) are introduced, and it is proved that (Formula presented.) - (Formula presented.) and (Formula presented.) - (Formula presented.) are equivalent. A detailed comparison with ideals, filters, grills, and primals highlights the distinctive features of the aura framework.