Complete metrizability of hyperspaces

Complete metrizability of the topology of bornological convergence on the hyperspace $\mathrm{CL}\left(X\right)$ of the nonempty closed subsets of a metric space $(X,d)$ is characterized. As a byproduct, it is shown under (CH), that the Attouch-Wets topology on $\mathrm{CL}\left(X\right)$ is completely metrizable if and only if the Hausdorff metric topology is, which is equivalent to $(X,d)$ being completely metrizable with a separable completion remainder $(\tilde{X}\setminus X,\tilde{d})$.