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

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?