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$hspace{1mm}$ B$-$Bourbaki-bounded homomorphisms. We compare them to each other and with the class of continuous homomorphisms. So, two topologies are presented on them and we determine 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 homomorphisms and B-compact homomorphisms briefly.