Generalized Open Sets and Closure Operators via Point-to-Neighborhood Assignments

We equip a topological space (X, τ) with a function a: X→τ satisfying the single axiom x∈a (x). The resulting triple (X, τ, a), 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 cla (A) =x∈X: a (x) ∩A≠⌀ is shown to be an additive Čech closure operator; it satisfies extensivity, monotonicity, and finite additivity, but idempotency fails in general. Iterating cla transfinitely yields a Kuratowski closure whose topology τa∞ satisfies τa∞=τa⊆τ, where τa is the collection of all a-open sets. We introduce a-semi-open, a-pre-open, a-α-open, and a-β-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 a-convergence of sequences and the corresponding continuity notions and their decompositions are studied. Separation axioms a-Ti (i = 0, 1, 2) are introduced, and it is proved that a-T1 and a-T2 are equivalent. A detailed comparison with ideals, filters, grills, and primals highlights the distinctive features of the aura framework.