This ain’t like dusting crops, boy

The hyperspace is a set of sets equipped with a metric or at least with a topology. Given a metric space X, let \mathcal{H}(X) be the set of all nonempty closed subsets of X with the Hausdorff metric: d(A,B)<r if no matter where you are in one set, you can jump into the other by traveling less than r. So, the distance between letters S and U is the length of the longer green arrow.

The requirement of closedness ensures d(A,B)>0 for A\ne B. If X is unbounded, then d(A,B) 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.

Imagine that

Every continuous surjection f\colon X\to Y has an inverse f^{-1}\colon Y\to \mathcal{H}(X) defined in the obvious way: f^{-1}(y)=f^{-1}(y). Yay ambiguous notation! The subset of \mathcal{H}(X) that consists of the singletons is naturally identified with X, so for bijective maps we recover the usual inverse.

Exercise: what conditions on f guarantee that f^{-1} is (a) continuous; (b) Lipschitz? After the previous post it should not be surprising that

  • Even if f is open and continuous, f^{-1} may be discontinuous.
  • If f is a Lipschitz quotient, then f^{-1} is Lipschitz.

Proofs are not like dusting crops—they are easier.

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