Fuzzy Aura Topological Spaces with Applications to Rough Set Theory and Medical Decision Making

TL;DR

This paper introduces fuzzy aura topological spaces, extending aura topology to fuzzy sets, and applies this framework to develop fuzzy rough set models and a new multi-criteria decision-making algorithm for medical diagnosis.

Contribution

It generalizes aura topological spaces to fuzzy settings, defines new operators and separation axioms, and proposes a fuzzy aura-based decision-making algorithm for medical applications.

Findings

Established hierarchy among fuzzy open set classes.

Developed fuzzy aura-closure and interior operators.

Applied the model to medical diagnosis with sensitivity analysis.

Abstract

We introduce the concept of a fuzzy aura topological space $(X, \tilde{\tau}, \tilde{a})$, obtained by equipping a Chang-type fuzzy topological space $(X, \tilde{\tau})$ with a fuzzy scope function $\tilde{a} : X \to \tilde{\tau}$ satisfying $\tilde{a}(x)(x) = 1$ for every $x \in X$. This framework generalizes the recently introduced (crisp) aura topological spaces to the fuzzy setting. We define the fuzzy aura-closure operator and the fuzzy aura-interior operator, and prove that the closure is a fuzzy additive \v{C}ech closure operator whose transfinite iteration yields a fuzzy Kuratowski closure. Five classes of generalized fuzzy open sets are introduced, and a complete hierarchy among them is established with counterexamples separating all distinct classes. Fuzzy aura-continuity and its decompositions are studied. Separation axioms and fuzzy aura-regularity are introduced. Fuzzy aura-based lower and upper approximation operators are defined, generalizing both the crisp aura rough set model and the Dubois-Prade fuzzy rough set model. A novel FA-MCDM algorithm is proposed and applied to a medical diagnosis problem with comprehensive sensitivity analysis.