Hyperspaces of two-dimensional continua

Michael Levin; Yaki Sternfeld

Fundamenta Mathematicae (1996)

  • Volume: 150, Issue: 1, page 17-24
  • ISSN: 0016-2736

Abstract

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Let X be a compact metric space and let C(X) denote the space of subcontinua of X with the Hausdorff metric. It is proved that every two-dimensional continuum X contains, for every n ≥ 1, a one-dimensional subcontinuum T n with d i m C ( T n ) ≥ n . This implies that X contains a compact one-dimensional subset T with dim C (T) = ∞.

How to cite

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Levin, Michael, and Sternfeld, Yaki. "Hyperspaces of two-dimensional continua." Fundamenta Mathematicae 150.1 (1996): 17-24. <http://eudml.org/doc/212159>.

@article{Levin1996,
abstract = {Let X be a compact metric space and let C(X) denote the space of subcontinua of X with the Hausdorff metric. It is proved that every two-dimensional continuum X contains, for every n ≥ 1, a one-dimensional subcontinuum $T_n$ with $dim C (T_n) ≥ n$. This implies that X contains a compact one-dimensional subset T with dim C (T) = ∞.},
author = {Levin, Michael, Sternfeld, Yaki},
journal = {Fundamenta Mathematicae},
keywords = {hyperspaces; hereditarily indecomposable continua; one- and two-dimensional continua; hereditarily indecomposable continuum},
language = {eng},
number = {1},
pages = {17-24},
title = {Hyperspaces of two-dimensional continua},
url = {http://eudml.org/doc/212159},
volume = {150},
year = {1996},
}

TY - JOUR
AU - Levin, Michael
AU - Sternfeld, Yaki
TI - Hyperspaces of two-dimensional continua
JO - Fundamenta Mathematicae
PY - 1996
VL - 150
IS - 1
SP - 17
EP - 24
AB - Let X be a compact metric space and let C(X) denote the space of subcontinua of X with the Hausdorff metric. It is proved that every two-dimensional continuum X contains, for every n ≥ 1, a one-dimensional subcontinuum $T_n$ with $dim C (T_n) ≥ n$. This implies that X contains a compact one-dimensional subset T with dim C (T) = ∞.
LA - eng
KW - hyperspaces; hereditarily indecomposable continua; one- and two-dimensional continua; hereditarily indecomposable continuum
UR - http://eudml.org/doc/212159
ER -

References

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  1. [1] R. H. Bing, Higher-dimensional hereditarily indecomposable continua, Trans. Amer. Math. Soc. 71 (1951), 267-273. Zbl0043.16901
  2. [2] A. Lelek, On mappings that change dimension of spheres, Colloq. Math. 10 (1963), 45-48. Zbl0113.16601
  3. [3] M. Levin, Hyperspaces and open monotone maps of hereditarily indecomposable continua, Proc. Amer. Math. Soc., to appear. Zbl0862.54011
  4. [4] M. Levin and Y. Sternfeld, Mappings which are stable with respect to the property dim f(X)≥ k, Topology Appl. 52 (1993), 241-265. 
  5. [5] M. Levin and Y. Sternfeld, The space of subcontinua of a 2-dimensional continuum is infinite dimensional, Proc. Amer. Math. Soc., to appear. Zbl0891.54004
  6. [6] S. B. Nadler Jr., Hyperspaces of Sets, Dekker, 1978. 
  7. [7] Y. Sternfeld, Mappings in dendrites and dimension, Houston J. Math. 19 (1993), 483-497. Zbl0819.54024

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