Properties of bipolar soft weak structures

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.