Abstract
Properties of the Hausdorff mapping H taking each compact metric space to the space of its non-empty closed subsets endowed with the Hausdorff metric are studied. It is shown that this mapping is non-stretching (i.e., a Lipschitz mapping whose Lipschitz constant equals 1). Several examples of classes of metric spaces such that the distances between them are preserved by the mapping H are given. The distance between any connected metric space with a finite diameter and any simplex with a greater diameter is calculated. Some properties of the Hausdorff mapping are discussed, which may be useful to understand whether the mapping H is isometric or not.
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