Growth hyperspaces of Peano continua

Abstract:

For X a nondegenerate Peano continuum, let ${2^X}$ be the hyperspace of all nonempty closed subsets of X, topologized with the Hausdorff metric. It is known that ${2^X}$ is homeomorphic to the Hilbert cube. A nonempty closed subspace $\mathcal {G}$ of ${2^X}$ is called a growth hyperspace provided it satisfies the following condition: if $A \in \mathcal {G}$, and $B \in {2^X}$ such that $B \supset A$ and each component of B meets A, then also $B \in \mathcal {G}$. The class of growth hyperspaces includes many previously considered subspaces of ${2^X}$. It is shown that if X contains no free arcs, and $\mathcal {G}$ is a nontrivial growth hyperspace, then $\mathcal {G}\backslash \{ X\}$ is a Hilbert cube manifold. A corollary characterizes those growth hyperspaces which are homeomorphic to the Hilbert cube. Analogous results are obtained for growth hyperspaces with respect to the hyperspace ${\text {cc}}(X)$ of closed convex subsets of a convex n-cell X.