Rough topological structures by various types of maximal neighborhoods

This manuscript centers on creating various topologies utilizing different sorts of maximal neighborhoods. The comparison of these topologies with the previous ones reveal that the earlier topology is weaker than the current ones. The core properties of the proposed topologies are examined, and the necessary conditions for achieving certain equivalences among them are outlined. Additionally, this study provides a distinctive characterization of these topologies by pinpointing the coarsest and largest one among all types, whereas previous methods were limited to characterizing only disjoint pairs of sets. Thereafter, these topologies are utilized to evolve new approximations. One of the major benefits of the current extension is that it adheres to all the properties of the original approximations without the constraints or limitations imposed by earlier versions. The significance of this paper lies not only in introducing new types of approximations based primarily on different kinds of topologies, but also in the fact that these approximations maintain the monotonic property for any given relation, enabling effective evaluation of uncertainty in the data. The monotonic property is crucial for various applications, as it guarantees that the approximation process is logically coherent and robust in the face of evolving information. The proposed models distinguish from their predecessors by their ability to compare all types of the suggested approximations. Moreover, comparisons reveal that the optimal approximations and accuracy are achieved with a specific type of generating topologies. The results demonstrate that topological notions can be a potent technique for studying rough set models. Furthermore, advanced topological features of approximate sets aid in finding rough measures, which assists in identifying missing feature values. Afterward, a numerical example is presented to highlight and emphasize the importance of the present results. Ultimately, the benefits of the followed manner are scrutinized and also some of their limitations are pointed out.