ARC-Smoothness and Contractibility in Whitney Levels

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.