Theoretical Advances and Improvements of "Generalized approximation spaces generation from $\mathbb{I}_{j}$-neighborhoods and ideals with application to Chikungunya disease" (European Journal of Pure and Applied Mathematics, 2025)
In their article, “Generalized approximation spaces generation from Ij -neighborhoods and ideals with application to Chikungunya disease,” published in AIMS Mathematics, 9(4) (2024), 10050–10077, T. M. Al-Shami and M. Hosny introduced a novel framework for constructing generalized approximation spaces utilizing Ij -neighborhoods and ideals. This current paper undertakes a critical validation of the published work. Our detailed investigation indicates that several core propositions and methodological instances require careful re-examination and precision, including the formulation of key results, underlying derivations, and illustrative examples. To address these areas, we provide rigorous counterexamples that highlight where further theoretical refinement is necessary. We subsequently correct and formally refine the established findings, offer enhanced mathematical formulations for critical concepts, and present new properties and clarifications. The goal of this paper is to strengthen the theoretical foundation of these generalized approximation spaces, ensuring maximal rigor and practical applicability for future research in the field.
Keywords: $\mathbb{I}_{j}$-neighborhoods; $\mathbb{I}^{\mathcal{K}}_{j}$-neighborhoods; approximation operators; rough sets; topology