Aura Topological Spaces and Generalized Open Sets with Applications to Rough Sets, Sensor Networks, and Epidemic Modelling

TL;DR

This paper introduces aura topological spaces, a new structure with a point-to-open-set assignment, exploring their properties and applications to rough sets, sensor networks, and epidemic modeling.

Contribution

It defines aura topological spaces, analyzes their closure operators and open sets, and applies these concepts to practical problems in data analysis and network modeling.

Findings

The aura-closure operator is additive but not idempotent.

Hierarchy of generalized open sets is fully characterized.

Applications include generalized rough set approximations, sensor coverage, and epidemic spread models.

Abstract

We equip a topological space $(X,\tau)$ with a function $\mathfrak{a}: X \to \tau$ satisfying the single axiom $x \in \mathfrak{a}(x)$. The resulting triple $(X, \tau, \mathfrak{a})$, 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 $\text{cl}_{\mathfrak{a}}(A) = \{x \in X : \mathfrak{a}(x) \cap A \neq \emptyset\}$ turns out to be an additive Cech closure operator; it satisfies extensivity, monotonicity, and finite additivity, but idempotency fails in general. Iterating $\text{cl}_{\mathfrak{a}}$ transfinitely yields a Kuratowski closure whose topology $\tau_{\mathfrak{a}}^{\infty}$ satisfies $\tau_{\mathfrak{a}}^{\infty} \subseteq \tau_{\mathfrak{a}} \subseteq \tau$. 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.