  In this paper, first we obtain some new and interesting results on projective
modules and on the upper topology of an ordinal number. Then it is shown that
the rank map of a locally of finite type projective module is continuous with
respect to the upper topology (by contract, it is well known this map is not
necessarily continuous with respect to the discrete topology). It is also
proved that a finitely generated flat module is projective if and only if its
rank map is continuous with respect to the upper topology.
