In this chapter we study a fuzzy topology and topological group with their separation axiom. A topological group is a mathematical structure that carries both a group and a topological space structure, with the two being compatible. Specifically, the group operations-multiplication and taking inverses-are required to be continuous with respect to the topology. The study of topological groups is significant both for its rich theoretical results and its wide range of applications, particularly in understanding continuous symmetries. Introducing an algebraic structure like a group to a topological space enhances its topological behavior and provides deeper insights. Topological groups, along with continuous group actions, play a crucial role in areas such as geometry, physics, and analysis.