# Movability and limits of polyhedra

Fundamenta Mathematicae (1993)

• Volume: 143, Issue: 3, page 191-201
• ISSN: 0016-2736

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## Abstract

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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 $\left({2}^{ℝ}2$, dS)$isseparable.Ontheotherhand,wegiveanexampleshowingthat$2ℝ2$isnotseparableinthefundamentalmetricintroducedbyBorsuk.$

## How to cite

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Laguna, V., et al. "Movability and limits of polyhedra." Fundamenta Mathematicae 143.3 (1993): 191-201. <http://eudml.org/doc/212003>.

@article{Laguna1993,
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 $(2^ℝ^2$, dS)$is separable. On the other hand, we give an example showing that$2ℝ2$is not separable in the fundamental metric introduced by Borsuk.$},
author = {Laguna, V., Moron, M., Nguyen, Nhu, Sanjurjo, J.},
journal = {Fundamenta Mathematicae},
keywords = {movability; polyhedron; shape metric; fundamental metric},
language = {eng},
number = {3},
pages = {191-201},
title = {Movability and limits of polyhedra},
url = {http://eudml.org/doc/212003},
volume = {143},
year = {1993},
}

TY - JOUR
AU - Laguna, V.
AU - Moron, M.
AU - Nguyen, Nhu
AU - Sanjurjo, J.
TI - Movability and limits of polyhedra
JO - Fundamenta Mathematicae
PY - 1993
VL - 143
IS - 3
SP - 191
EP - 201
AB - 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 $(2^ℝ^2$, dS)$is separable. On the other hand, we give an example showing that$2ℝ2$is not separable in the fundamental metric introduced by Borsuk.$
LA - eng
KW - movability; polyhedron; shape metric; fundamental metric
UR - http://eudml.org/doc/212003
ER -

## References

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11. [11] J. Dydak and J. Segal, Theory of Shape : An Introduction, Lecture Notes in Math. 688, Springer, Berlin, 1978. Zbl0401.54028
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14. [14] V. Klee, Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30-45. Zbl0064.10505
15. [15] S. Mardešić and J. Segal, Shape Theory, North-Holland, 1982.
16. [16] S. B. Nadler, Hyperspaces of Sets, Dekker, New York, 1978. Zbl0432.54007
17. [17] H. Noguchi, A generalization of absolute neighbourhood retracts, Kodai Math. Sem. Rep. 1 (1953), 20-22. Zbl0052.18803
18. [18] S. Spież, Movability and uniform movability, Bull. Acad. Polon. Sci. 22 (1974), 43-45. Zbl0279.54021
19. [19] J. H. Wells and L. R. Williams, Embeddings and Extensions in Analysis, Springer, Berlin, 1975. Zbl0324.46034

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