# 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) 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.

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.