2026/02/07 by Ahu Acikgoz · 1 voice
Mathematics · #math.GN
We equip a topological space (X,τ) with a function \mathfraka: X → τ satisfying the single axiom x ∈ \mathfraka(x). The resulting triple (X, τ, \mathfraka), which we call an aura topological space, provides a point-to-open-set assignment that differs from all existing auxiliary structures in topology. The aura-closure operator cl_\mathfraka(A) = \x ∈ X : \mathfraka(x) ∩ A ≠ ∅\ turns out to be an additive Cech closure operator; it satisfies extensivity, monotonicity, and finite additivity, but idempotency fails in general. Iterating cl_\mathfraka transfinitely yields a Kuratowski closure whose topology τ_\mathfraka∞ satisfies τ_\mathfraka∞ ⊆ τ_\mathfraka ⊆ τ. We introduce five classes of generalized open sets, determine their complete hierarchy, and separate all non-coinciding classes by counterexamples. Continuity notions, decomposition theorems, and separation axioms are studied. Three applications are developed: rough set approximations generalizing Pawlak's model, wireless sensor network coverage analysis, and epidemic spread modelling.