Movability and limits of polyhedra

Abstract

We define a metric d(S), called the shape metric, on the hyperspace 2X of all non-empty compact subsets of a metric space X. Using it we prove that a compactum X in the Hilbert cube is movable if and only if X is the limit of a sequence of polyhedra in the shape metric. This fact is applied to show that the hyperspace (2R2, d(S)) is separable. On the other hand, we give an example showing that 2R2 is not separable in the fundamental metric introduced by Borsuk.