On a characterization of complete metrizability of the Hausdorff metric topology

It is shown, under CH, that the Hausdorff metric topology on the hyperspace $CL(X)$ of all nonempty closed subsets of a metric space $(X,d)$ is completely metrizable if and only if $(X,d)$ is completely metrizable and its completion remainder $(\tilde X\setminus X,\tilde d)$ is separable.