  We show that all non-trivial continuous endomorphisms of the circle group are
topologically mixing. We also show that there exists a large infinite class of
continuous endomorphisms of any n-dimensional torus group which are
topologically mixing.
  Lastly, we prove that any continuous endomorphism on an abelian polish
semigroup (with an identity) can be extended in a natural way to a
topologically mixing endomorphism on the countable infinite product of said
semigroup. This shows that every countable infinite product of an abelian
polish semigroup has a topologically mixing endomorphism and, in particular,
the countable infinite toroidal group has infinitely many topologically mixing
endomorphisms.
