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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

A $ 2$-sphere in $ E\sp{3}$ with vertically connected interior is tame


Authors: J. W. Cannon and L. D. Loveland
Journal: Trans. Amer. Math. Soc. 195 (1974), 345-355
MSC: Primary 57A10
MathSciNet review: 0343273
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Abstract: A set X in $ {E^3}$ is said to have vertical number n if the intersection of each vertical line with X contains at most n components. The set X is said to have vertical order n if each vertical line intersects X in at most n points. A set with vertical number 1 is said to be vertically connected. We prove that a 2-sphere in $ {E^3}$ with vertically connected interior is tame. This result implies as corollaries several previously known taming theorems involving vertical order and vertical number along with several more general and previously unknown results.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0343273-5
PII: S 0002-9947(1974)0343273-5
Keywords: Taming sets, $ \ast $-taming sets, vertical order, vertical number, vertically connected, 2-spheres in $ {E^3}$, surfaces in 3-manifolds
Article copyright: © Copyright 1974 American Mathematical Society