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

摘要

The space clos( X ) of all nonempty closed subsets of an unbounded metric space X is considered. The space 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 θ are bounded and, for any r , the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on 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 clos( X ) are obtained; criteria for the bounded compactness and separability of clos( X ) are given. The space of continuous maps from a compact space to clos( X ) is considered; conditions for a set to be totally bounded in this space are found.