Growth hyperspaces of Peano continua

Abstract

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