Movability and limits of polyhedra
Volume 143 / 1993
Fundamenta Mathematicae 143 (1993), 191-201
DOI: 10.4064/fm-143-3-191-201
Abstract
We define a metric $d_S$, called the shape metric, on the hyperspace $2^X$ 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 $(2^{ℝ^2}, d_S)$ is separable. On the other hand, we give an example showing that $2^{ℝ^2}$ is not separable in the fundamental metric introduced by Borsuk.
