Soft aura topological spaces and rough approximation operators

Abstract

We introduce the concept of a soft aura topological space $(X, \tildeτ, \mathfrak{a}_E)$, obtained by equipping a soft topological space $(X, \tildeτ, E)$ with a soft scope function $\mathfrak{a}_E : X \to \tildeτ$ satisfying $x \in \mathfrak{a}_E(x)(e)$ for every $x \in X$ and every parameter $e \in E$. This framework generalizes the recently introduced aura topological spaces to the soft setting. We define the soft aura-closure operator and the soft aura-interior operator, and prove that the closure is a soft additive Čech closure operator whose transfinite iteration yields a soft Kuratowski closure. Five classes of generalized soft open sets -- soft $\mathfrak{a}$-semi-open, soft $\mathfrak{a}$-pre-open, soft $\mathfrak{a}$-$α$-open, soft $\mathfrak{a}$-$β$-open, and soft $\mathfrak{a}$-$b$-open sets -- are introduced, and a complete hierarchy among them is established. Soft $\mathfrak{a}$-continuity and its decompositions are studied. Separation axioms soft $\mathfrak{a}$-$T_i$ ($i = 0, 1, 2, 3$) are introduced; it is shown that soft $\mathfrak{a}$-$T_1$ and soft $\mathfrak{a}$-$T_2$ coincide due to the scope-based formulation. Soft aura-based lower and upper rough approximation operators are defined, generalizing both the crisp aura rough set model and the classical Pawlak model. An illustrative application to environmental risk assessment demonstrates the practical utility of the proposed framework.

Theorems & Definitions (56)

• Definition 3.1
• Definition 3.2
• Example 3.1
• Remark 3.1
• Remark 3.2
• Definition 3.3
• Definition 3.4
• Theorem 3.1
• proof
• Remark 3.3