Ideal-Aura Topological Spaces, New Local Functions, and Generalized Open Sets
Ahu Acikgoz
TL;DR
The paper develops the ideal-aura framework by combining an ideal topological space with a scope function to define the aura-local function $A^{\mathfrak{a}}(\mathcal{I})$ and the Čech closure $\operatorname{cl}^{*}_{\mathfrak{a}}$, establishing a topology chain $\tau_{\mathfrak{a}} \subseteq \tau^{*}_{\mathfrak{a}} \subseteq \tau^{*}$ whose idempotency is equivalent to the transitivity of $\mathfrak{a}$. It introduces the $\psi_{\mathfrak{a}}$-operator and a basis $\beta_{\mathfrak{a}}(\mathcal{I})$ to describe $\tau^{*}_{\mathfrak{a}}$, and defines five hierarchical classes of $\mathcal{I}\mathfrak{a}$-generalized open sets with decomposition results for $\mathcal{I}\mathfrak{a}$-continuity. The framework recovers known cases at endpoints (pure aura, discrete topology) and reveals localization phenomena for intermediate ideals, while offering a suite of open problems related to stabilization ordinals, product behavior, and possible extensions with codomain ideals. Overall, the work unifies ideal-topological enrichments with aura-geometry to yield a rich, interpolating topological structure and a structured theory of generalized continuity.
Abstract
We combine an ideal topological space $(X, τ, \mathcal{I})$ with a scope function $\mathfrak{a}: X \to τ$, $x \in \mathfrak{a}(x)$, to form what we call an ideal-aura topological space $(X, τ, \mathcal{I}, \mathfrak{a})$. The central new object is the aura-local function $A^{\mathfrak{a}}(\mathcal{I}) = \{x \in X : \mathfrak{a}(x) \cap A \notin \mathcal{I}\}$, which extends the Jankovic-Hamlett local function: we always have $A^{*}(\mathcal{I}, τ) \subseteq A^{\mathfrak{a}}(\mathcal{I})$. The closure $\operatorname{cl}^{*}_{\mathfrak{a}}(A) = A \cup A^{\mathfrak{a}}(\mathcal{I})$ is an additive Cech closure operator that, in general, fails to be idempotent; we prove that idempotency is equivalent to transitivity of $\mathfrak{a}$. The resulting Cech topology $τ^{*}_{\mathfrak{a}}$ sits in the chain $τ_{\mathfrak{a}} \subseteq τ^{*}_{\mathfrak{a}} \subseteq τ^{*}$, interpolating between the pure aura topology and the classical ideal topology. We introduce a $ψ_{\mathfrak{a}}$-operator and use it to give an alternative description of $τ^{*}_{\mathfrak{a}}$. Five classes of $\mathcal{I}\mathfrak{a}$-generalized open sets are defined and arranged in a hierarchy, with strict inclusions separated by counterexamples. Decomposition theorems for $\mathcal{I}\mathfrak{a}$-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.