This paper introduces and investigates a new class of weak structures called bipolar soft weak structures within the framework of bipolar soft topological spaces. It then explores their core properties within bipolar soft set systems. Some topological notions such as bipolar soft weak open, bipolar soft weak closed, bipolar soft weak closure, bipolar soft weak interior, and bipolar soft weak boundary sets are introduced and analyzed, along with their interrelations. Further examines bipolar soft weak neighborhoods, and bipolar soft weak limit points. Additionally, the concepts of bipolar soft weak subspace is presented. Moreover, it provides illustrative examples, remarks, theorems, and propositions to support the definitions and properties discussed. In addition, a real-world application is presented to demonstrate the practical efficiency of the proposed structure. A qualitative computational complexity analysis is conducted to show its operational cost and scalability. Furthermore, a comparative discussion among the soft weak structure, bipolar soft topology, and the proposed bipolar soft weak structure highlights their axiomatic and operational distinctions.

The rapid advancement of smart healthcare systems demands sophisticated mathematical tools to manage uncertainty and conflicting expert opinions in critical decision-making (DM) processes. Multi-criteria group decision making (MCGDM) plays a pivotal role in synthesizing diverse expert evaluations for complex healthcare challenges. However, existing soft set (SS) extensions often struggle to simultaneously capture multinary evaluations, bipolar reasoning, higher-order fuzzy logic, and multi-expert input. To overcome these limitations, we propose the Fermatean fuzzy N-bipolar soft expert set (FFNBSES), which enhances fuzzy representation while integrating multinary, bipolar, and multi-expert evaluations. We formally define the fundamental operations of FFNBSES and demonstrate its algebraic properties. A DM methodology based on FFNBSES is developed and applied to a healthcare case study, showcasing its superior capability to handle expert consensus and disagreement in multi-criteria evaluations. Comparative analysis within the SS theory framework highlights the enhanced flexibility and robustness of FFNBSES for real-world MCGDM problems. This work provides a powerful and comprehensive approach to support smart healthcare transformation through improved group DM under uncertainty.

The purpose of this paper is to define bipolar soft generalized compact sets and bipolar soft generalized compact spaces. The structures of $\widetilde{\stackrel{~}{g}}$-centralized bipolar soft generalized closed sets collection in a bipolar soft generalized compact space are given. Moreover, some main properties of bipolar soft generalized compactness are discussed and their relationships are studied. The concept of a bipolar soft generalized compactness is introduced and it investigates under what condition a bipolar soft generalized topological space forms a bipolar soft generalized compact space. The relation between bipolar soft generalized compact space and soft generalized compact space is proposed. Furthermore, some further properties of bipolar soft mappings, such as bipolar soft composite mappings, are presented and some of their characteristics are explained. Additionally, novel classes of bipolar soft mapping such as bipolar soft generalized continuous, bipolar soft generalized open, and bipolar soft generalized closed mappings are defined. Finally, some results and counterexamples are obtained.