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Arc-smoothness and contractibility in Whitney levels


Author: Alejandro Illanes
Journal: Proc. Amer. Math. Soc. 110 (1990), 1069-1074
MSC: Primary 54F15; Secondary 54B20
DOI: https://doi.org/10.1090/S0002-9939-1990-1037210-2
MathSciNet review: 1037210
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Abstract: Let $ X$ be a continuum. Let $ {2^x}$ (resp., $ C(X)$) be the space of all nonempty closed subsets (resp., subcontinua) of $ X$. In this paper we prove that if $ X$ is an arc-smooth continuum, then there exists an admissible Whitney map $ \mu :{2^x} \to {\mathbf{R}}$ such that $ \mu \vert C(X):C(X) \to {\mathbf{R}}$ is admissible and for every $ t \in (0,\mu (X)),{\mu ^{ - 1}}(t)$ and $ {(\mu \vert C(X))^{ - 1}}(t)$ are arc-smooth. This answers a question by J. T. Goodykoontz, Jr. Also we give an example of a contractible continuum $ X$ such that, for every Whitney map $ \upsilon :C(X) \to {\mathbf{R}}$ there exists $ t \in (0,\upsilon (X))$ such that $ {\upsilon ^{ - 1}}(t)$ is not contractible.


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  • [1] J. B. Fugate, G. R. Gordh, Jr., and L. Lum, On arc-smooth continua, Topology Proc. 2 (1977), 645-656.
  • [2] -, Arc-smooth continua, Trans. Amer. Math. Soc. 265 (1981), 545-561. MR 610965 (82j:54072)
  • [3] J. T. Goodykoontz, Jr., Arc-smoothness in hyperspaces, Topology Appl. 15 (1985), 131-150. MR 686091 (85a:54012)
  • [4] J. T. Goodykoontz, Jr. and S. B. Nadler, Jr., Whitney levels in hyperspaces of certain Peano continua, Trans. Amer. Math. Soc. 274 (1982), 671-694. MR 675074 (84h:54010)
  • [5] J. Krasinkiewicz and S. B. Nadler, Jr., Whitney properties, Fund. Math. 98 (1978), 165-180. MR 0467691 (57:7546)
  • [6] S. B. Nadler, Jr., Hyperspaces of sets, Pure Appl. Math., vol. 49, Dekker, New York, 1978. MR 0500811 (58:18330)
  • [7] A. Petrus, Contractibility of Whitney continua in $ C(X)$, Gen. Topology Appl. 9 (1978), 275-288. MR 510909 (80a:54010)

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DOI: https://doi.org/10.1090/S0002-9939-1990-1037210-2
Article copyright: © Copyright 1990 American Mathematical Society

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