  We show that the set of entries generated by any finite set of doubly
stochastic matrices is nowhere dense, in contrast to the cases of stochastic
matrices or unitary matrices. In other words, there is no finite universal set
of doubly stochastic matrices, even with the weakest notion of universality.
  Our proof is based on a theorem for topological semigroups with the
convergent property. A topological semigroup is convergent if every infinite
product converges. We show that in a compact and convergent semigroup, under
some restrictions, the closure of every finitely generated subsemigroup can be
instead generated directly by the generating set and the limits of infinite
products.
