Boundedness and continuity in topological groups

This thesis covers three chapters such a way that in the first chapter, to makeunderstood other chapters easily we give place to topological space, topologicalvector spaces and basic concepts in topological group.In the second chapter, boundedness concept for topological groups is presentedand association of boundedness with topological concepts such as, compactness,precompactness and connectedness has been analyzed.Moreover, defined d (x; y) = p (x¡1y) with the help of absolute value functionp and features of semi metric d that produce one group topology over group Ghave been analyzed. Bounded transformation concept through topological groups isdefined and results of bounded homomorphism features for topological groups andbornological groups have been obtained. Moreover, definition and theorems relatedto characterized bounded sets by invariant pseudo-metric or metrics and topologicalgroups into homomorphic transformation have been given.In the last chapter, Pontryagin duality concept in topological groups is defined.The conspicuous point of chapter is to give an example of weak (Bohr) bounded setin topological group would not be always bounded in original topology.KEY WORDS: Topological Groups, boundedness, bounded homomorphisms,Pontryagin duality, Bornological Groups.