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Given a continuum with a metric , we let to denote the
*hyperspace* of all nonempty closed subsets of equipped with the
*Hausdorff metric* defined by
(see e.g. [Nadler 1978, (0.1), p. 1 and (0.12), p. 10]).
If tends
to zero as tends to infinity, we put
. Further, we denote
by the hyperspace of singletons of , and by the hyperspace of all
subcontinua of , i.e., of all connected elements of . Since is
homeomorphic to , there is a natural embedding of into ,
and so we can write
. Thus one can consider a
retraction from either or onto .

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*Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih*

*2001-11-30*