A Hilbert space limit for the iterated hyperspace functor

Abstract:

Let $X$ be a nondegenerate metric Peano continuum and let $P(X)$ be the hyperspace of closed, nonvoid subsets of $X$ equipped with the Hausdorff metric. Then the inclusion of $X$ into $P(X)$ as the single element sets is an isometry and we have a direct system $X \to P(X) \to P(P(X)) \to \cdots$ of isometric inclusions. Let $X’$ be the metric direct limit and ${X^*}$ be its completion. We prove that the pair $({X^*},X’)$ is homeomorphic to the pair $({l^2},l_\sigma ^2)$, where $l_\sigma ^2$ is the linear span in ${l^2}$ of the Hilbert cube.