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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Some universally pierced arcs in $ E\sp{3}$


Author: L. D. Loveland
Journal: Proc. Amer. Math. Soc. 49 (1975), 469-474
MSC: Primary 57A10
MathSciNet review: 0370590
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Abstract: A subset $ X$ of $ {E^3}$ is said to be universally pierced if each $ 2$-sphere containing $ X$ can be pierced by a tame arc at each point of $ X$. We show that an arc $ A$ is universally pierced provided $ A$ has a shrinking point $ p$ such that either $ p$ lies in a tame arc in $ A$ or $ {E^3} - A$ has $ 1$-ALG at $ p$. Applying this result we show the existence of infinitely many wild universally pierced arcs.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0370590-1
PII: S 0002-9939(1975)0370590-1
Keywords: Piercing points, pierced sets, universally pierced, tame and wild arcs, surfaces in $ {E^3}$, shrinking points, $ 1$-ALG sets, cellular arcs
Article copyright: © Copyright 1975 American Mathematical Society