Ideal-Aura Topological Spaces, New Local Functions, and Generalized Open Sets

TL;DR

This paper introduces ideal-aura topological spaces, extending local functions and generalized open sets, and explores their properties, hierarchies, and special cases, bridging classical and new topological concepts.

Contribution

It defines ideal-aura topological spaces and associated generalized open sets, establishing new closure operators, topologies, and decomposition theorems, advancing the theory of generalized topological structures.

Findings

The closure operator $ ext{cl}^{*}_{ ext{a}}$ is additive but not always idempotent.

Idempotency of the closure operator is equivalent to transitivity of $ ext{a}$.

Hierarchies of generalized open sets are established with strict inclusions.

Abstract

We combine an ideal topological space $(X, \tau, \mathcal{I})$ with a scope function $\mathfrak{a}: X \to \tau$, $x \in \mathfrak{a}(x)$, to form what we call an ideal-aura topological space $(X, \tau, \mathcal{I}, \mathfrak{a})$. 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 $A^{*}(\mathcal{I}, \tau) \subseteq A^{\mathfrak{a}}(\mathcal{I})$. 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 $\tau^{*}_{\mathfrak{a}}$ sits in the chain $\tau_{\mathfrak{a}} \subseteq \tau^{*}_{\mathfrak{a}} \subseteq \tau^{*}$, interpolating between the pure aura topology and the classical ideal topology. We introduce a $\psi_{\mathfrak{a}}$-operator and use it to give an alternative description of $\tau^{*}_{\mathfrak{a}}$. 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.