  A subbundle of variable dimension inside the tangent bundle of a smooth
manifold is called a smooth distribution if it is the pointwise span of a
family of smooth vector fields. We prove that all such distributions are
finitely generated, meaning that the family may be taken to be a finite
collection. Further, we show that the space of smooth sections of such
distributions need not be finitely generated as a module over the smooth
functions. Our results are valid in greater generality, where the tangent
bundle may be replaced by an arbitrary vector bundle.
