  We show that the dimension of the Cuntz semigroup of a C*-algebra is
determined by the dimensions of the Cuntz semigroups of its separable
sub-C*-algebras. This allows us to remove separability assumptions from
previous results on the dimension of Cuntz semigroups.
  To obtain these results, we introduce a notion of approximation for abstract
Cuntz semigroups that is compatible with the approximation of a C*-algebra by
sub-C*-algebras. We show that many properties for Cuntz semigroups are
preserved by approximation and satisfy a L\"owenheim-Skolem condition.
