The Cuntz semigroup of continuous fields Ramon AntoineJoan BosaFrancesc Perera 46L3546L8006F05Cuntz semigroupC*-algebrascontinuous fieldsclassification In this paper, we describe the Cuntz semigroup of continuous fields of $\mathrm{C}^{*}$-algebras over one-dimensional spaces whose fibers have stable rank one and trivial $K_1$ 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. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5071 10.1512/iumj.2013.62.5071 en Indiana Univ. Math. J. 62 (2013) 1105 - 1131 state-of-the-art mathematics http://iumj.org/access/