An Introduction to Topological groups

Abstract

This thesis introduces the theory of Topological groups highlighting their fundamental properties and mathematical significance. after a brief overview of key concepts from topology and algebra in chapter 1 . the core of the thesis lies in chapter 2 where we define Topological groups and study their main structural features. these include the continuity of group operations , neighborhood systems at the identity , subgroups , quotient topological groups and the uniform structure associated with topological groups. chapter 3 outlines selected applications in harmonic analysis (including Haar measure and Pontryagin duality ) lie Theory and functional analysis with further examples from physics and computer science.