P-ORDER ANALYSIS OF e-OPEN SETS IN CUBIC PICTURE FUZZY TOPOLOGICAL SPACES

P-ORDER ANALYSIS OF e-OPEN SETS IN CUBIC PICTURE FUZZY TOPOLOGICAL SPACES

G. Saravanakumar, K. Suganthi

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Abstract

In this paper, we introduce and investigate new classes of generalized open sets-namely, e-open, e∗-open, a-open, and β-open sets—within the framework of Cubic Picture Fuzzy Topological Spaces (CPFTS’s) under P-order. Our study extends the theory of generalized open sets to the richer structure of CPFTS’s, which combine interval-valued and tri-component fuzzy membership functions. We provide formal definitions for these open sets in CPFTS’s and explore their fundamental topological properties. We establish a series of characterizations and inclusion relationships among the defined sets, supported by illustrative examples and counterexamples. These results highlight the distinct nature of each open set type and their non-reversible implications under set-theoretic operations. Our approach emphasizes both algebraic and topological aspects, revealing a refined hierarchy of openness in the context of CPFTS’s. Furthermore, we analyze the behavior of these sets under P-order, offering insights into how order structures influence topological properties in fuzzy environments. The dual-order framework enables a deeper understanding of neighborhood systems and proximity relations in CPFTS. This study fills an important gap in the literature by assimilating generalized open set theory with CPFTS’s for the first time. The findings pave the way for future research in fuzzy topology and its applications in decision-making systems, intelligent environments, and uncertainty modeling.

Keywords

Cubic Picture Fuzzy Topological Spaces, e-Open Sets, P-Order.