Paper's Title:

R-Order Analysis of e-Open Continuous Mapping in Cubic Picture Fuzzy Topological Spaces

Author(s):

G. Saravanakumar and K. Suganthi

Department of Mathematics,
Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology (Deemed to be University),
Avadi, Chennai-600062,
India.
E-mail: saravananguru2612@gmail.com

 Department of Mathematics,
Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology (Deemed to be University),
Avadi, Chennai-600062,
India.
E-mail: suganthiselvam0980@gmail.com 

Abstract:

This paper investigates the structure and properties of {e-continuous mappings} within the framework of Cubic Picture Fuzzy Topological Spaces (CPFTSs) under R-order. Building on the foundational role of e-open sets in generalized topological structures, we introduce several classes of continuity-namely e-continuous, δP-continuous, δS-continuous a-continuous, β-continuous, and e*-continuous maps -- and study their interrelationships through logical implications and set-theoretic operations. We establish necessary and sufficient conditions for each type of continuity, along with non-reversible inclusion results supported by counterexamples. Furthermore, we present preservation results involving closure and interior operations under e-continuous maps and identify conditions under which e-continuity implies classical R-order continuity. The findings contribute to the refinement of topological structures in fuzzy environments and offer a foundational basis for functional modeling under uncertainty, with applications in decision theory, AI systems, and fuzzy information modeling.

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