Ps- Group Spaces :A Novel Approach in Topology

This study introduces a novel theoretical framework known as Ps-Group Spaces, which integrates topology and algebra in a generalized context. The research builds upon the development of Ps-Open Sets, an extension of semi-open sets, which are utilized to examine key topological properties such as Ps-Compactness and Ps-Hausdorff Spaces. Through rigorous formalization, Ps-Group Spaces are defined as a unifying structure that bridges algebraic and topological frameworks, providing a new approach to exploring non-conventional spaces. The research demonstrates the compatibility of Ps-Group Spaces with compactness and separation properties, highlighting their unique advantages over traditional topological groups. These spaces are shown to address complex problems in dynamic systems, particularly in systems with irregular boundaries or weak separation properties. Practical applications include analyzing physical systems such as fluid dynamics, enhancing classification algorithms and dimensionality reduction in artificial intelligence, and studying the stability of open markets like cryptocurrency systems. The research concludes with suggestions for future studies, focusing on applications in algebraic geometry and complex physical systems.