On the Metric Space of Closed Subsets of a Metric Space and Set-Valued Maps with Closed Images

The space $\operatorname{clos}(X)$ of all nonempty closed subsets of an unbounded metric space $X$ is considered. The space $\operatorname{clos}(X)$ is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point $\theta$ are bounded and, for any $r$, the sequence of the unions of the given sets with the exterior balls of radius $r$ centered at $\theta$ converges in the Hausdorff metric. The metric on $\operatorname{clos}(X)$ thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space $X$. Conditions for a set to be closed, totally bounded, or compact in $\operatorname{clos}(X)$ are obtained; criteria for the bounded compactness and separability of $\operatorname{clos}(X)$ are given. The space of continuous maps from a compact space to $\operatorname{clos}(X)$ is considered; conditions for a set to be totally bounded in this space are found.