Some examples of monothetic groups

In this paper we discuss in Section 1 the finest group topology on Z zuch that {2n}nN is a sequence converging to zero. We prove that Z is a complete topological group with respect to that topology. The proof is almost entirely self-contained. In section 2 we give an example of a monothetic group (that is a topological group which contains a dense copy of Z) which is complete metrizable, totally disconnected and which has no continuous characters except 0. The example is constructed by starting with a complete meterizable totally disconnected monothetic group and factoring out a discrete subgroup which is dense in the Bohr compactification. The main difficulty here is in proving discreteness. Section 2 relies heaviliy on a trick developed in Section 1, namely the use of a kind of "coordinates" for 2-adic numbers. Section 0 contains a survey of related results already existing in the literature.