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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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