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$^\ast$-Taming Sets for Crumpled Cubes. II: Horizontal Sections in Closed Sets

James W. Cannon
Transactions of the American Mathematical Society
Vol. 161 (Nov., 1971), pp. 441-446
DOI: 10.2307/1995951
Stable URL: http://www.jstor.org/stable/1995951
Page Count: 6
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
$^\ast$-Taming Sets for Crumpled Cubes. II: Horizontal Sections in Closed Sets
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Abstract

We prove that a closed subset $X$ of $E^3$ is a $^\ast$-taming set if no horizontal section of $X$ has a degenerate component. This implies, for example,that a 2-sphere $S$ in $E^3$ is tame if no horizontal section of $S$ has a degenerate component. It also implies (less obviously) that a 2-sphere S in $E^3$ is tame if it can be touched at each point from each side of $S$ by a pencil.

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