This paper introduces N-hypersoft (N-HS) sets—an enriched and versatile extension of hypersoft (HS) sets—designed to handle evaluations involving both binary and non-binary data while embodying an inherent sense of structural symmetry. The paper presents several algebraic definitions, including incomplete N-HS sets, efficient N-HS sets, normalized N-HS sets, equivalence under normalization, N-HS complements, and HS sets derived from a threshold. These definitions are accompanied by illustrative examples. Additionally, the paper delves into various set-theoretic operations within the framework of N-HS sets, such as relative null/whole N-HS sets, N-HS subsets, and N-HS extended/restricted union and intersection, presented in two different ways. Finally, the paper presents and compares decision-making methodologies regarding N-HS sets.

The basic of bipolar soft set theory stands for a mathematical instrument that brings together the soft set theory and bipolarity. Its definition is based on two soft sets, a set that provides positive information and other that gives negative. This paper mainly aims at defining a new bipolar soft generalized topological space; setting out of the point that the collection of bipolar soft sets forms the basis for the definition of the new concept is defined. Added to that, an investigation has been made of the four concepts of bipolar soft generalized, namely g-interior, g-closure, g-exterior and g-boundary. Furthermore, the main properties of bipolar soft generalized topological space (BSGT S) are established. This paper also attends to the discussion of the relations between these new definitions and the application of the given bipolar soft generalized topological spaces in a decision-making problem where an algorithm for this application has been suggested. Finally, to clarify and substantiate what the current work subsumes, some examples have been provided.

In this paper, we apply the concept of soft sets to δ-open set and δ-closed set. The associated soft δ-topology in terms of soft δ-open sets were introduced and some properties of them were investigated. Moreover, the definitions, characterizations and basic results concerning soft δ-interior, soft δ-closure, soft δ-boundary and soft δ-exterior were given. Finally, the concept of soft pu−δ- continuity was defined and some properties of it were introduced.

Multiple criteria decision-making (MCDM) is a significant area of decision-making theory that certainly warrants attention. It might be difficult to accurately convey the necessary decision facts when navigating decision-making problems since we frequently run into complicated issues and unpredictable situations. To address this, introducing the novel idea of the bipolar pqr-spherical fuzzy set (${\mathrm{B}}_{\mathrm{pqr}}$SFS), a hybrid structure of the bipolar fuzzy set (BFS) and the pqr-spherical fuzzy set (pqr-SFS), is the main goal of this work. The fundamental (set-theoretic and algebraic) operations on ${\mathrm{B}}_{\mathrm{pqr}}$SFSs are explained as well as their relations to several known models. A distance measure, such as Euclidean distance, among ${\mathrm{B}}_{\mathrm{pqr}}$SFNs, is provided. Afterward, we expand the fundamental aggregation operators to the pqr-spherical fuzzy (${\mathrm{B}}_{\mathrm{pqr}}$SF) environment by developing bipolar pqr-spherical fuzzy-weighted averaging and bipolar pqr-spherical fuzzy-weighted geometric operators for aggregating ${\mathrm{B}}_{\mathrm{pqr}}$SFNs. According to the aforementioned distance measure and operators, an MCDM approach is established consisting of two algorithms, namely, the TOPSIS method and the method using the proposed operators in the ${\mathrm{B}}_{\mathrm{pqr}}$SF context. Moreover, a numerical example is provided in order to ensure that the presented model is applicable. By using the two algorithms, a comparative analysis of the proposed method with other existing ones is given in order to verify the feasibility of the suggested decision-making procedure.

The objective of studing the current paper is to introduced a new class of soft open sets in soft topological spaces called soft -open sets. Then soft -open sets are used to study some soft topological concepts. Furthermore, the concept of soft -continuous and almost soft -continuous functions are defined by using the soft -open sets. Some properties and Characterizations of such functions are given.

The paper develops a novel analysis of mutual interactions between topology and soft topology. It is known that each soft topology produces a system of crisp (parameterized) topologies. The other way round is also possible. Namely, one can generate a soft topology from a system of crisp topologies. Different methods of producing soft topologies are discussed by implementing two formulas. Then, the relationships between the resulting soft topologies are obtained. With the help of an example, it is demonstrated that one formula is more constructible than the other. Now, it is reasonable to ask which (topological) properties of a soft topology can be transferred to the set of crisp topologies, or the opposite. To address this question, we consider the standard separation axioms and show how well these axioms can be preserved when moving from a system of crisp topologies to the soft topology generated by it and contrariwise. Additionally, our findings extend and disprove some results from the literature.