Gradient approximation of vector fields

In this paper, we consider an approximation problem which arose in optimal control theory. We seek conditions on a compact subset K of Euclidean n-space such that every continuous vector field on K may be uniformly approximated (on K) by vector fields with strictly positive integrating factors. We prove that such approximation is possible for all K in a particular subclass of the compact sets with topological dimension less than or equal to 1.