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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Whitney levels in hyperspaces of certain Peano continua

Authors: Jack T. Goodykoontz and Sam B. Nadler
Journal: Trans. Amer. Math. Soc. 274 (1982), 671-694
MSC: Primary 54B20; Secondary 54C99, 54F20
MathSciNet review: 675074
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Abstract: Let $ X$ be a Peano continuum. Let $ {2^x}$ (resp., $ C(X)$) be the space of all nonempty compacta (resp., subcontinua) of $ X$ with the Hausdorff matric. Let $ \omega$ be a Whitney map defined on $ \mathcal{H}={2^{X}}$ or $ C(X)$ such that $ \omega$ is admissible (this requires the existence of a certain type of deformation of $ \mathcal{H}$). If $ \mathcal{H}=C(X)$, assume $ X$ contains no free arc. Then, for any $ {t_0} \in (0,\omega (X))$, it is proved that $ {\omega ^{ - 1}}({t_0}),\,{\omega ^{ - 1}}([0,\,{t_0}])$, and $ {\omega ^{ - 1}}([{t_0},\,\omega (X)])$ are Hilbert cubes. This is an analogue of the Curtis-Schori theorem for $ \mathcal{H}$. A general result for the existance of admissible Whitney maps is proved which implies that these maps exist when $ X$ is starshaped in a Banach space or when $ X$ is a dendrite. Using these results it is shown, for example that being an AR, an ANR, an LC space, or an $ {\text{L}}{{\text{C}}^n}$ space is not strongly Whitney-reversible.

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Keywords: Absolute retract, acyclicity, admissible Whitney map, arc-smooth, cell-like map, contractible, dendrite, dendroid, Hilbert cube, hyperspace, locally $ n$-connected, Peano continuum, shape, smooth dendroid, starshaped, strong Whitney-reversible property, Whitney level, Whitney map, Whitney stable, $ Z$-set
Article copyright: © Copyright 1982 American Mathematical Society