The hyperspace is a set of sets equipped with a metric or at least with a topology. Given a metric space , let
be the set of all nonempty closed subsets of
with the Hausdorff metric:
if no matter where you are in one set, you can jump into the other by traveling less than
. So, the distance between letters S and U is the length of the longer green arrow.
The requirement of closedness ensures for
. If
is unbounded, then
will be infinite for some pairs of sets, which is natural: the hyperspace contains infinitely many parallel universes which do not interact, being at infinite distance from one another.

Every continuous surjection has an inverse
defined in the obvious way:
. Yay ambiguous notation! The subset of
that consists of the singletons is naturally identified with
, so for bijective maps we recover the usual inverse.
Exercise: what conditions on guarantee that
is (a) continuous; (b) Lipschitz? After the previous post it should not be surprising that
- Even if
is open and continuous,
may be discontinuous.
- If
is a Lipschitz quotient, then
is Lipschitz.
Proofs are not like dusting crops—they are easier.