Isometries between a metric space and its hyperspace, function space, and space of measures

摘要

It is shown that 1. there does not exist a (bounded) metric space X with more than one element which is isometric to the metric space CL(X) of nonempty and closed subsets of X endowed with the Hausdorff metric, and 2. there does not exist a (bounded) metric space X which is isometric to the metric space N(X) of nonexpansive functions from X to R endowed with the supremum metric. Whether 3. there exists a (bounded) metric space X with more than one element which is isometric to the metric space M(X) of Borel probability measures on X endowed with the Hutchinson metric is still open.