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It is well-known that the law of a one-dimensional diffusion on natural scaleis fully characterized by its speed measure. C. Stone proved a continuousdependence of diffusions on their speed measures. In this paper we establishthe converse direction, i.e. we prove a continuous dependence of the speedmeasures on their diffusions. Furthermore, we take a topological point of viewon the relation. More precisely, for suitable topologies, we establish ahomeomorphic relation between the set of regular diffusions on natural scalewithout absorbing boundaries and the set of locally finite speed measures.

On the relation of one-dimensional diffusions on natural scale and their speed measures