There is a close analogy and similarity between topology and rough set theory. As, the leading idea of this theory is depended on two approximations, namely lower and upper approximations, which correspond to the interior and closure operators in topology, respectively. So, the joined study of this theory and topology becomes fundamental. This theory mainly propose to enlarge the lower approximations by adding new elements to it, which is an equivalent goal for canceling elements from the upper approximations. For this intention, one of the primary motivation of this paper is the desire of improving the accuracy measure and reducing the boundary region. This aim can be achieved easily by utilizing ideal in the construction of the approximations as it plays an important role in removing the vagueness of concept. The emergence of ideal in this theory leads to increase the lower approximations and decrease the upper approximations. Consequently, it minimizes the boundary and makes the accuracy higher than the previous. Therefore, this work expresses the set of approximations by using new topological notions relies on ideals namely $\mathcal{I}$- ${{\delta }_{\beta }}_J$-open sets and $\mathcal{I}$- ${\underset{\beta }{\bigwedge }}_J$-sets. Moreover, these notions are also utilized to extend the definitions of the rough membership relations and functions. The essential properties of the suggested approximations, relations and functions are studied. Comparisons between the current and previous studies are presented and turned out to be more precise and general. The brilliant idea of these results is increased in importance by applying it in the chemical field as it is shown in the end of this paper. Additionally, a practical example induced from an information system is introduced to elucidate that the current rough membership functions is better than the former ones in the other studies.

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.