Fermatean Double-Valued Neutrosophic Soft Topological Spaces

This paper introduces the concept of Fermatean Double Valued Neutrosophic Soft Set (FDVNSS), an advanced mathematical framework for modeling uncertainty, vagueness, and indeterminacy in complex systems. A FDVNSS extends traditional neutrosophic and soft set theories by incorporating two layers of truth and falsity degrees: absolute and relative. This multi-dimensional representation allows for more nuanced decision-making, particularly in domains characterized by conflicting or imprecise information, such as medical diagnostics. The formal definition of FDVNSS is established, including the conditions under which membership degrees satisfy the Fermatean constraint, i.e., the sum of the cubes of the degrees lies within a defined range. Key operations such as subset, null and absolute FDVNSS, complement, union, and intersection are rigorously defined and exemplified. Furthermore, the algebraic properties of FDVNSSs under these operations are explored, and foundational propositions-analogous to classical set theory- are proven to hold. Examples drawn from health care contexts, such as evaluating diseases, symptoms, or medical responses, are provided to illustrate the applicability of FDVNSS in realworld problems. Overall, this framework provides a more expressive and flexible tool for reasoning under high levels of uncertainty and offers significant potential for application in intelligent decision support systems. The notion of a Fermatean Double Valued Neutrosophic Soft Topological Space (FDVNSTS) is introduced. The concepts of semi-open (s-open), pre-open (p-open), and b-open sets are defined within the context of FDVNSTS. Among these generalized open sets, the pre-open set is selected for further exploration, and several fundamental topological notions are developed based on this definition. These include the closure, exterior, boundary, and interior in FDVNSTS.