Exploring diverse topological rough group structures: results and applications

Various topological rough group structures, such as S-topological rough groups, s-topological rough groups, Irr-topological rough groups, and irresolute topological rough groups, naturally arise in numerous topological and geometrical contexts. Additionally, subgroups within these proposed structures are introduced and examined along with their properties. This work formulates fundamental definitions and results, including examples, counterexamples, and an exploration of their interrelations. The main objective is to advance the development of a comprehensive theory for these diverse topological rough groups and to inspire further exploration of their applications.