  In the present paper we investigate the properties of the Hausdorff mapping
$\mathcal{H}$, which takes each compact metric space to the space of its
nonempty closed subspaces. It is shown that this mapping is nonexpanding
(Lipschitz mapping with constant $1$). This paper gives several examples of
classes of metric spaces, distances between which are preserved by the mapping
$\mathcal{H}$. We also calculate distance between any connected metric space
and any simplex with greater diameter than the former one. At the end of the
paper we discuss some properties of the Hausdorff mapping which may help to
prove that it is isometric
