A Rigorous Treatment of Supra Feebly Open Sets in Supra Topological Spaces

Abstract

This work presents a systematic revision and rigorous extension of the classical theory on supra feebly-open sets within supra topological spaces. We formally define a supra feebly-open (s.f.o.) subset A by the presence of a supra open set U∈ w such that U⊆A⊆scl_si^w​(U) and we rigorously establish the equivalent formulation A⊆ cl^wsi(Int^w (A)). The manuscript explores the fundamental properties of these sets, presents the supra feebly interior and supra feebly closures operators, and meticulously charts defines their inclusion connections with supra α-open, supra semi-open, supra pre-open, supra b-open, and supra β-open sets. This study enhances logical consistency, standardizes notation and puts the results in context of the overall generalized weak openness. The theoretical advantages  provide strong useful foundation for future topological applications in the areas of digital topology, rough set theory and fuzzy/neutrosophic supra structures. Mathematical consistency, logical coherence, and formal scholarly presentation have been meticulously ensured throughout the presentation to provide future topological uses.