New Class of Ideal Topological Spaces

In this study, we establish a newly constructed operators for establishing a new kind of sets entitled  sets.  By preserving both local and transitional qualities in a topological space, these sets generalize and improve a number of traditional and generalized topological constructions.  We examine the structural properties of  sets in comparison to pre-open forms as well as semi-regular closed sets. We show that the  set represents strictly weaker compared to the α-open sets in general.  Additionally, we demonstrate that all  sets constitute a supratopology by showing that their collection occurs under arbitrary unions. The newly developed family of sets offers fresh insights into continuity, closure, and convergent analysis as well as a fundamental basis for creating sophisticated ideas in extended topological spaces. These results offer a theoretical structure that may be expanded to examine intricate connections between different generalized open sets, opening up new avenues for sophisticated topological modeling applications.Furthermore, we demonstrate that the collection of all -K sets forms a supratopology, as it is closed under arbitrary unions. Theintroduction This is a new family of sets.provides fresh perspectives on continuity, closure operations, and convergence analysis, offering a robust framework for developing advanced notions in extended topological structures. The findings presented in this work open new avenues for exploring intricate relationships among various generalized open sets and pave the way for sophisticated modeling applications in modern topology.