A -sphere in with vertically connected interior is tame

Authors:
J. W. Cannon and L. D. Loveland

Journal:
Trans. Amer. Math. Soc. **195** (1974), 345-355

MSC:
Primary 57A10

MathSciNet review:
0343273

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Abstract | References | Similar Articles | Additional Information

Abstract: A set *X* in is said to have *vertical number n* if the intersection of each vertical line with *X* contains at most *n* components. The set *X* is said to have *vertical order n* if each vertical line intersects *X* in at most *n* points. A set with vertical number 1 is said to be *vertically connected*. We prove that a 2-sphere in with vertically connected interior is tame. This result implies as corollaries several previously known taming theorems involving vertical order and vertical number along with several more general and previously unknown results.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0343273-5

Keywords:
Taming sets,
-taming sets,
vertical order,
vertical number,
vertically connected,
2-spheres in ,
surfaces in 3-manifolds

Article copyright:
© Copyright 1974
American Mathematical Society