The Cuntz semigroup of continuous fields

In this paper, we describe the Cuntz semigroup of continuous fields of C*-algebras over one-dimensional spaces whose fibers have stable rank one and trivial K1 for each closed, two-sided ideal. This is done in terms of the semigroup of global sections on a certain topological space built out of the Cuntz semigroups of the fibers of the continuous field. Furthermore, when the fibers have real rank zero, and when we take into account the action of the space, our description yields that the Cuntz semigroup is a classifying invariant if and only if the sheaf is also induced by the Murray-von Neumann semigroup.