ON SOME WEAKER REGULAR AND NORMAL FORMS OF ALMOST TOPOLOGICAL SPACE UNDER THE CONTINUOUS MAPPING

Abstract In 1971, Čirić introduced the concept of nearly topological spaces, which minimizes topological constraints while keeping key properties, this study examines this generality. We examine why poorer nearly topological spaces affect continuity, convergence, compactness, and connectedness, we want to develop new mathematical frameworks for functional analysis, approximation theory, and fixed-point theory by exploring how axiom and open set definition modifications impact these structures, we found that weaker structures may broaden topological applications beyond mathematics, we also examine how weaker forms impact mappings and continuous transformations. New tools and approaches for interdisciplinary mathematics and related fields are the goal of this program, this study proves that continuous mappings to other spaces need topological space regularity and normalcy, it addresses how virtually topological spaces enhance classical topology and affect dynamical systems, algebraic topology, and functional analysis, studies on these areas enhance topology and related fields, we focus on the effects of weak topological forms in nearly topological spaces, particularly the function \( f \) and its importance in compact regular spaces, this study details reduced regularity and normalcy to understand their relations in essentially topological spaces, the findings enhance topological categorization, provide the framework for future research, and influence mathematical modelling and theoretical computer science by establishing structural stability under continuous mappings, notable contributions, virtual topological space theory advances, understanding the complex relationships between compactness, regularity, and normalcy.