2026/02/07 by Ahu Acikgoz, Murad Ozkoc · 1 voice · 1 citation
Mathematics · #math.GN
arxiv published 2026/02/07 · arxiv updated 2026/02/16
We combine an ideal topological space (X, τ, I) with a scope function \mathfraka: X → τ, x ∈ \mathfraka(x), to form what we call an ideal-aura topological space (X, τ, I, \mathfraka). The central new object is the aura-local function A^\mathfraka(I) = \x ∈ X : \mathfraka(x) ∩ A ∉ I\, which extends the Jankovic-Hamlett local function: we always have A*(I, τ) ⊆ A^\mathfraka(I). The closure cl*_\mathfraka(A) = A ∪ A^\mathfraka(I) is an additive Cech closure operator that, in general, fails to be idempotent; we prove that idempotency is equivalent to transitivity of \mathfraka. The resulting Cech topology τ*_\mathfraka sits in the chain τ_\mathfraka ⊆ τ*_\mathfraka ⊆ τ*, interpolating between the pure aura topology and the classical ideal topology. We introduce a ψ_\mathfraka-operator and use it to give an alternative description of τ*_\mathfraka. Five classes of I\mathfraka-generalized open sets are defined and arranged in a hierarchy, with strict inclusions separated by counterexamples. Decomposition theorems for I\mathfraka-continuity are proved. Three special cases are examined: the trivial ideal recovers the pure aura topology, the improper ideal gives the discrete topology, and the ideal of finite sets exhibits a localization phenomenon.