A Topological Framework For Approximation Spaces Induce By Finite Group Conjugacy

Abstract

This paper introduces a novel theoretical framework for constructing topological approximation spaces by leveraging the algebraic structure of finite groups. The core innovation lies in utilizing the conjugacy relation (⋆R) as a fundamental equivalence relation to partition the universal set U. These conjugacy classes serve as the base for generating a specific topology (⋆τ) on the group. Within this hybrid environment, a new type of approximation is established based on topological interior and closure operators. The study demonstrates significant results, proving that every base element in this topology is a clopen set. Furthermore, it is established that for Abelian groups, the induced topology (⋆τ) consistently results in a discrete topology. The analysis is further extended to define a topological rough membership function (  effectively bridging algebraic properties with rough set concepts to handle uncertain information. To validate the efficiency of this framework, comprehensive analytical examples are provided using the Dihedral group (D₃) and the Quaternion group (Q₈ ).