  This paper studies topological definitions of chain recurrence and shadowing
for continuous endomorphisms of topological groups generalizing the relevant
concepts for metric spaces. It is proved that in this case the sets of chain
recurrent points and chain transitive component of the identity are topological
subgroups. Furthermore, it is obtained that some dynamical properties induced
by the original system on quotient spaces. These results link an algebraic
property to a dynamical property.
