Fundamental Theorems and Some Properties of Maximal 〖iD〗_g^#-Open Sets in quad topological spaces

This study deals with the fundamental properties of maximal -open sets in field of topology. The focus is on a set of important theoretical results, including the proof of the decomposition theorem, which explains how these sets can be divided into simpler subsets while preserving their intrinsic properties. In addition, the study investigates the properties associated with the intersection of maximal -open sets, with an emphasis on relationship between these intersections and the low of radical closure. Exact proofs of these properties are provided and their theoretical importance in understanding the dynamics of open sets in this type is demonstrated. The outcomes contribute to the mathematical understanding of topology by providing new analysis methods that can be applied to the study of a new class of open sets.