Arc-smoothness and contractibility in Whitney levels
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- by Alejandro Illanes PDF
- Proc. Amer. Math. Soc. 110 (1990), 1069-1074 Request permission
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 $\mu |C(X):C(X) \to {\mathbf {R}}$ is admissible and for every $t \in (0,\mu (X)),{\mu ^{ - 1}}(t)$ and ${(\mu |C(X))^{ - 1}}(t)$ 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.References
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Additional Information
- © Copyright 1990 American Mathematical Society
- 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