A -sphere of vetical order bounds a -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 of is said to have vertical order if no vertical line contains more than points of . We prove that each -sphere in which has vertical order 5 bounds a -cell.

**[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*, Rocky Mt. J. Math. (to appear).**[4]**J. W. Cannon,*-taming sets for crumpled cubes*. I:*Basic properties*, (to appear). MR**0282353 (43:8065)****[5]**-,*-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*, 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)**

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 -spheres,
tame surfaces,
embeddings in ,
surfaces in ,
-taming sets,
vertical order

Article copyright:
© Copyright 1970
American Mathematical Society