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 Löwenheim-Skolem condition.