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

Fundamenta Mathematicae (1992)

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

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topIllanes, 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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- [2] S. Eilenberg, Transformations continues en circonférence et la topologie du plan, Fund. Math. 26 (1936), 61-112. Zbl0013.42002
- [3] A. Illanes, Arc-smoothness and contractibility in Whitney levels, Proc. Amer. Math. Soc. 110 (1990), 1069-1074. Zbl0707.54008
- [4] A. Illanes, Spaces of Whitney maps, Pacific J. Math. 139 (1989), 67-77. Zbl0674.54012
- [5] A. Illanes, The space of Whitney levels, Topology Appl., to appear. Zbl0761.54010
- [6] A. Illanes, Spaces of Whitney decompositions, An. Inst. Mat. Univ. Nac. Autónoma México 28 (1988), 47-61.
- [7] A. Illanes, The space of Whitney levels is homeomorphic to ${l}_{2}$, Colloq. Math., to appear. Zbl0819.54016
- [8] A. Illanes, Arc smoothness is not a Whitney reversible property, Aportaciones Mat.: Comun. 8 (1990), 65-80.
- [9] J. Krasinkiewicz and S. B. Nadler, Jr., Whitney properties, Fund. Math. 98 (1978), 165-180.
- [10] S. Mardešić, Equivalence of singular and Čech homology for ANR-s. Application to unicoherence, ibid. 46 (1958), 29-45. Zbl0085.37303
- [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] S. B. Nadler, Hyperspaces of Sets, Dekker, New York 1978. Zbl0432.54007
- [13] A. Petrus, Contractibility of Whitney continua in C(X), Gen. Topology Appl. 9 (1978), 275-288. Zbl0405.54006
- [14] L. E. Ward, Jr., Extending Whitney maps, Pacific J. Math. 93 (1981), 465-469. Zbl0457.54008

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