  We say that a C*-algebra is nowhere scattered if none of its quotients
contains a minimal open projection. We characterize this property in various
ways, by topological properties of the spectrum, by divisibility properties in
the Cuntz semigroup, by the existence of Haar unitaries for states, and by the
absence of nonzero ideal-quotients that are elementary, scattered or type I.
  Under the additional assumption of real rank zero or stable rank one, we show
that nowhere scatteredness implies even stronger divisibility properties of the
Cuntz semigroup.
