Risk-controlled neutrosophic nano closed set separation axioms in neutrosophic nano topological spaces with decision-making applications

Uncertainty and indeterminacy are inherent in many real-world decision-making problems, where classical and fuzzy topological models often prove insufficient to capture risk-sensitive information. To address this limitation, this paper introduces a Risk-Controlled Neutrosophic Nano Closed \((RCNNC\) ) set framework within neutrosophic nano topological spaces. By incorporating a risk threshold based on indeterminacy and falsity components, the proposed approach provides a refined mechanism for handling uncertainty in a controlled manner. New \(RCNNC\) separation axioms, namely \(RCNNC\) \({T₀},{T₁}\) and \({T₂}\) , are formulated and their structural properties and hierarchical relationships are investigated. Formal theorems with rigorous proofs are established to demonstrate the consistency and distinctiveness of the proposed axioms. Furthermore, the applicability of the \(RCNNC\) framework is illustrated through decision-making models, including a medical diagnosis decision support system, where alternatives are classified into safe, uncertain, and high-risk regions. The proposed model effectively bridges theoretical neutrosophic nano topology and practical decision-making under uncertainty, offering a robust and interpretable foundation for future research.