Topological manifestations of group-theoretic structures

Abstract

Inspired by approximation operators in rough set theory and utilizing group theoretical concepts, approximation operators on groups have been defined in this study. These operators have been used to construct various topological structures. Key topologies introduced on groups include the cn-topology derived from cyclic subgroups, the Cn-topology from centralizers, the Nn-topology from normalizers, the Ln-topology and Rn-topology from left and right cosets of a subgroup, and the on-topology from group actions. It has been determined that isomorphic groups yield homeomorphic spaces, revealing a fundamental connection between group structures and topological properties. Furthermore, it has been observed that for Abelian groups, the cn-topology, Ln-topology, and Rn-topology form topological groups.