A new characterization of tame -spheres in

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 is tame from if and only if for each compact set there exists a 2-sphere with complementary domains , such that and for each there exists a path in of diameter less than which runs from *x* to a point . Furthermore, the theorem holds when *A* is replaced by *B*, by by , 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 , *S* has a metric -envelope in *C* which is a 2-sphere. Theorem 3 answers affirmatively the following question: Is a 2-sphere tame in if there exists an such that if *a*, satisfy , then there exists a path in *S* of spherical diameter which connects *a* and *b*?

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0339188-9

Keywords:
Tame surfaces,
tame 2-spheres,
surfaces in ,
characterizations of tameness

Article copyright:
© Copyright 1974
American Mathematical Society