Movability and limits of polyhedra

Volume 143 / 1993

V. F. Laguna, M. A. Moron, Nguyen To Nhu, J. M. R. Sanjurjo Fundamenta Mathematicae 143 (1993), 191-201 DOI: 10.4064/fm-143-3-191-201


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.


  • V. F. LagunaDepartamento de Matematica Fundamental
    Facultad De Ciencias, U.N.E.D.
    28040 Madrid, Spain
  • M. A. MoronDepartamento de Matematicas
    E.T.S.I. de Montes
    Universidad Politecnica
    28040 Madrid, Spain
  • Nguyen To NhuInstitute of Mathematics
    P.O. Box 631
    Bo Ho
    Hanoi, Vietnam
  • J. M. R. SanjurjoDepartamento De Geometria y Topologia
    Facultad de Matematicas
    Universidad Complutense
    28040 Madrid, Spain

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