  We introduce a notion of covering dimension for Cuntz semigroups of
C*-algebras. This dimension is always bounded by the nuclear dimension of the
C*-algebra, and for subhomogeneous C*-algebras both dimensions agree.
  Cuntz semigroups of Z-stable C*-algebras have dimension at most one. Further,
the Cuntz semigroup of a simple, Z-stable C*-algebra is zero-dimensional if and
only if the C*-algebra has real rank zero or is stably projectionless.
