Set-valued metrics and generalized Hausdorff distances

Let $X$ be a metric space and $BCl(X)$ the collection of nonempty bounded closed subsets of $X$. We show that Hausdorff distance $d_H$ belongs to a specific family of real-valued distances on $BCl(X)$, each of which can be expressed as the composition $\mu\circ d_{sv}$ of a topology inducing set-valued function $d_{sv}:BCl(X)^2\rightarrow \mathcal{P}(Z)$ and a real-valued set-function $\mu:\Sigma\subset\mathcal{P}(Z)\rightarrow\mathbb{R}$. With this observation, we construct several associated classes of inter-set distances, called set-valued metrics and generalized Hausdorff distances. Our constructions are both explicit and adaptable, and the resulting distance classes are expected to cover most practical applications involving distance between sets.