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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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A new characterization of tame $2$-spheres in $E^{3}$
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by Lawrence R. Weill PDF
Trans. Amer. Math. Soc. 190 (1974), 243-252 Request permission

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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Additional Information
  • © Copyright 1974 American Mathematical Society
  • 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