We give a description of the closure of the natural a ne contin-
uous function representation of K0(R) for any exchange ring R. This goal is
achieved by extending the results of Goodearl and Handelman, about metric
completions of dimension groups, to a more general class of pre-ordered groups,
which includes K0 of exchange rings. As a consequence, the results about K+
0
of regular rings, which the author gave in an earlier paper, can be extended
to a wider class of rings, which includes C -algebras of real rank zero, among
others. Also, the framework of pre-ordered groups developed here allows other
potential applications
