A characterization of dendroids by the n-connectedness of the Whitney levels

Alejandro Illanes

Fundamenta Mathematicae (1992)

  • Volume: 140, Issue: 2, page 157-174
  • ISSN: 0016-2736

Abstract

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Let X be a continuum. Let C(X) denote the hyperspace of all subcontinua of X. In this paper we prove that the following assertions are equivalent: (a) X is a dendroid, (b) each positive Whitney level in C(X) is 2-connected, and (c) each positive Whitney level in C(X) is ∞-connected (n-connected for each n ≥ 0).

How to cite

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Illanes, Alejandro. "A characterization of dendroids by the n-connectedness of the Whitney levels." Fundamenta Mathematicae 140.2 (1992): 157-174. <http://eudml.org/doc/211935>.

@article{Illanes1992,
abstract = {Let X be a continuum. Let C(X) denote the hyperspace of all subcontinua of X. In this paper we prove that the following assertions are equivalent: (a) X is a dendroid, (b) each positive Whitney level in C(X) is 2-connected, and (c) each positive Whitney level in C(X) is ∞-connected (n-connected for each n ≥ 0).},
author = {Illanes, Alejandro},
journal = {Fundamenta Mathematicae},
keywords = {Whitney map; Whitney property; continuum; dendroid; Whitney level},
language = {eng},
number = {2},
pages = {157-174},
title = {A characterization of dendroids by the n-connectedness of the Whitney levels},
url = {http://eudml.org/doc/211935},
volume = {140},
year = {1992},
}

TY - JOUR
AU - Illanes, Alejandro
TI - A characterization of dendroids by the n-connectedness of the Whitney levels
JO - Fundamenta Mathematicae
PY - 1992
VL - 140
IS - 2
SP - 157
EP - 174
AB - Let X be a continuum. Let C(X) denote the hyperspace of all subcontinua of X. In this paper we prove that the following assertions are equivalent: (a) X is a dendroid, (b) each positive Whitney level in C(X) is 2-connected, and (c) each positive Whitney level in C(X) is ∞-connected (n-connected for each n ≥ 0).
LA - eng
KW - Whitney map; Whitney property; continuum; dendroid; Whitney level
UR - http://eudml.org/doc/211935
ER -

References

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  1. [1] C. Eberhart and S. B. Nadler, Jr., The dimension of certain hyperspaces, Bull. Acad. Polon. Sci. 19 (1971), 1027-1034. Zbl0235.54037
  2. [2] S. Eilenberg, Transformations continues en circonférence et la topologie du plan, Fund. Math. 26 (1936), 61-112. Zbl0013.42002
  3. [3] A. Illanes, Arc-smoothness and contractibility in Whitney levels, Proc. Amer. Math. Soc. 110 (1990), 1069-1074. Zbl0707.54008
  4. [4] A. Illanes, Spaces of Whitney maps, Pacific J. Math. 139 (1989), 67-77. Zbl0674.54012
  5. [5] A. Illanes, The space of Whitney levels, Topology Appl., to appear. Zbl0761.54010
  6. [6] A. Illanes, Spaces of Whitney decompositions, An. Inst. Mat. Univ. Nac. Autónoma México 28 (1988), 47-61. 
  7. [7] A. Illanes, The space of Whitney levels is homeomorphic to l 2 , Colloq. Math., to appear. Zbl0819.54016
  8. [8] A. Illanes, Arc smoothness is not a Whitney reversible property, Aportaciones Mat.: Comun. 8 (1990), 65-80. 
  9. [9] J. Krasinkiewicz and S. B. Nadler, Jr., Whitney properties, Fund. Math. 98 (1978), 165-180. 
  10. [10] S. Mardešić, Equivalence of singular and Čech homology for ANR-s. Application to unicoherence, ibid. 46 (1958), 29-45. Zbl0085.37303
  11. [11] S. B. Nadler, Jr., Some basic connectivity properties of Whitney maps inverses in C(X), in: Studies in Topology, Proc. Charlotte Topology Conference (University of North Carolina at Charlotte, 1974), Academic Press, New York 1975, 393-410. 
  12. [12] S. B. Nadler, Hyperspaces of Sets, Dekker, New York 1978. Zbl0432.54007
  13. [13] A. Petrus, Contractibility of Whitney continua in C(X), Gen. Topology Appl. 9 (1978), 275-288. Zbl0405.54006
  14. [14] L. E. Ward, Jr., Extending Whitney maps, Pacific J. Math. 93 (1981), 465-469. Zbl0457.54008

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