Homomorphisms on Topological Groups from the Perspective of Bourbaki-boundedness

In this note we study some topological properties of bounded sets and Bourbaki-bounded sets. Also we introduce two types of Bourbaki- bounded homomorphisms on topological groups including, n􀀀Bourbaki- bounded homomorphisms and B􀀀Bourbaki-bounded homomorphisms. We compare them to each other and with the class of continuous ho- momorphisms. So, two topologies are presented on them and we de- termine some properties on domain and range spaces led to Bourbaki- completeness some of these classes of homomorphisms with the given topologies. At the end of this note we focus on n-compact homomor- phisms and B-compact homomorphisms brie y.