Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A $ 2$-sphere of vetical order $ 5$ bounds a $ 3$-cell


Author: L. D. Loveland
Journal: Proc. Amer. Math. Soc. 26 (1970), 674-678
MSC: Primary 54.78
DOI: https://doi.org/10.1090/S0002-9939-1970-0268871-0
MathSciNet review: 0268871
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A subset $ X$ of $ {E^3}$ is said to have vertical order $ n$ if no vertical line contains more than $ n$ points of $ X$. We prove that each $ 2$-sphere in $ {E^3}$ which has vertical order 5 bounds a $ 3$-cell.


References [Enhancements On Off] (What's this?)

  • [1] J. W. Alexander, An example of a simply connected surface bounding a region which is not simply connected, Proc. Nat. Acad. Sci. U. S. A. 10 (1924), 8-10.
  • [2] R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 456-483. MR 19, 300. MR 0087090 (19:300f)
  • [3] C. E. Burgess and J. W. Cannon, Embeddings of surfaces in $ {E^3}$, Rocky Mt. J. Math. (to appear).
  • [4] J. W. Cannon, $ \ast $-taming sets for crumpled cubes. I: Basic properties, (to appear). MR 0282353 (43:8065)
  • [5] -, $ \ast $-taming sets for crumpled cubes. III: Horizontal sections in closed sets, (to appear).
  • [6] R. H. Fox and E. Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. (2) 49 (1948), 979-990. MR 10, 317. MR 0027512 (10:317g)
  • [7] R. A. Jensen and L. D. Loveland, Surfaces of vertical order 3 are tame, Bull. Amer. Math. Soc. 76 (1970), 151-154. MR 0250281 (40:3520)
  • [8] L. D. Loveland, Tame subsets of spheres in $ {E^3}$, Pacific J. Math. 19 (1966), 489-517. MR 37 #903. MR 0225309 (37:903)
  • [9] -, Piercing points of crumpled cubes, Trans. Amer. Math. Soc. 143 (1969), 145-152. MR 0247619 (40:883)
  • [10] D. R. McMillan, Jr., Some topological properties of piercing points, Pacific J. Math. 22 (1967), 313-322. MR 35 #7319. MR 0216486 (35:7319)
  • [11] R. L. Moore, Foundations of point set theory, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 13, Amer. Math. Soc., Providence, R. I., 1962. MR 0150722 (27:709)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54.78

Retrieve articles in all journals with MSC: 54.78


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0268871-0
Keywords: Tame $ 2$-spheres, tame surfaces, embeddings in $ {E^3}$, surfaces in $ {E^3}$, $ \ast $-taming sets, vertical order
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society