The notion of the radical of a topological Abelian group can be found in N. Bourbaki [Eléments de mathématique, Livre III, Topologie générale (1947; Zbl 0030.24102), §2, Ex. 28]. A real character of a topological Abelian group is a continuous homomorphism from it into the group of all real numbers. It is shown that a connected topological Abelian group has sufficiently many real characters if and only if it is radical-free, i.e. its radical is equal to zero. For a locally compact Abelian group $$G$$ it is proved that $$G$$ is connected (totally disconnected) if and only if its dual group $$G^*$$ is radical-free (radical). It is proved that the radical of a locally compact connected group is a topological direct summand of it.