Set-valued metrics and generalized Hausdorff distances

Abstract: Let $X$ be a (topological) space and $Cl(X)$ the collection of nonempty closed subsets of $X$. Given a topology on $Cl(X)$, making $Cl(X)$ a space, a \emph{(subset) hyperspace} of $X$ is any subspace $\mathcal{J}\subset Cl(X)$ with an embedding $X\hookrightarrow\mathcal{J}$, $x\mapsto{x}$ (which thus requires $X$ to be $T_1$). In this note, we highlight a key attribute of the Hausdorff distance $d_H$ on $Cl(X)$, namely, \emph{the expressibility of $d_H$ as the composition of a set-valued function and a real-valued set-function}. Using this attribute of $d_H$, we describe associated classes of distances called \emph{set-valued metrics} and \emph{generalized Hausdorff distances}.