  The Vietoris space of compact subsets of a given Hausdorff space yields an
endofunctor $\mathscr V$ on the category of Hausdorff spaces. Vietoris
polynomial endofunctors on that category are built from $\mathscr V$, the
identity and constant functors by forming products, coproducts and
compositions. These functors are known to have terminal coalgebras and we
deduce that they also have initial algebras. We present an analogous class of
endofunctors on the category of extended metric spaces, using in lieu of
$\mathscr V$ the Hausdorff functor $\mathcal H$. We prove that the ensuing
Hausdorff polynomial functors have terminal coalgebras and initial algebras.
Whereas the canonical constructions of terminal coalgebras for Vietoris
polynomial functors takes $\omega$ steps, one needs $\omega + \omega$ steps in
general for Hausdorff ones. We also give a new proof that the closed set
functor on metric spaces has no fixed points.
