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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A new characterization of tame $ 2$-spheres in $ E\sp{3}$


Author: Lawrence R. Weill
Journal: Trans. Amer. Math. Soc. 190 (1974), 243-252
MSC: Primary 57A50; Secondary 57A10
DOI: https://doi.org/10.1090/S0002-9947-1974-0339188-9
MathSciNet review: 0339188
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Abstract: It is shown in Theorem 1 that a 2-sphere S in $ {E^3}$ is tame from $ A = {\text{Int}}\;S$ if and only if for each compact set $ F \subset A$ there exists a 2-sphere $ S'$ with complementary domains $ A' = {\text{Int}}\;S',B' = {\text{Ext}}\;S'$, such that $ F \subset A' \subset \overline {A'} \subset A$ and for each $ x \in S'$ there exists a path in $ \overline {B'} $ of diameter less than $ \rho (F,S)$ which runs from x to a point $ y \in S$. Furthermore, the theorem holds when A is replaced by B, $ A'$ by $ B',B'$ by $ A'$, and Int by Ext. Two applications of this characterization are given. Theorem 2 states that a 2-sphere is tame from the complementary domain C if for arbitrarily small $ \varepsilon > 0$, S has a metric $ \varepsilon $-envelope in C which is a 2-sphere. Theorem 3 answers affirmatively the following question: Is a 2-sphere $ S \subset {E^3}$ tame in $ {E^3}$ if there exists an $ \varepsilon > 0$ such that if a, $ b \in S$ satisfy $ \rho (a,b) < \varepsilon $, then there exists a path in S of spherical diameter $ \rho (a,b)$ which connects a and b?


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DOI: https://doi.org/10.1090/S0002-9947-1974-0339188-9
Keywords: Tame surfaces, tame 2-spheres, surfaces in $ {E^{3}}$, characterizations of tameness
Article copyright: © Copyright 1974 American Mathematical Society