  It is well-known that the law of a one-dimensional diffusion on natural scale
is fully characterized by its speed measure. C. Stone proved a continuous
dependence of diffusions on their speed measures. In this paper we establish
the converse direction, i.e. we prove a continuous dependence of the speed
measures on their diffusions. Furthermore, we take a topological point of view
on the relation. More precisely, for suitable topologies, we establish a
homeomorphic relation between the set of regular diffusions on natural scale
without absorbing boundaries and the set of locally finite speed measures.
