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A Note about Locally Spherical Spheres

Published online by Cambridge University Press:  20 November 2018

W. T. Eaton
W. T. Eaton*
Affiliation:
The University of Tennessee, Knoxville, Tennessee
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A 2-sphere S in E3 is said to be locally spherical if for each point p in S and each ∈ > 0 there is a 2-sphere S' such that p Ç Int S”, S' ᴖ S is a continuum, and Diam S' < ∈. It is not known whether locally spherical spheres are tame; however, there are several partial results. Burgess (2) showed that S is tame if S' C\ S is a simple closed curve and Loveland (3) proved that S is tame if S can be side approximated missing the continuum S ᴖ S'. In this paper we demonstrate that S is tame if the continuum S P\ S' is irreducible with respect to separating S. This result is stated more precisely in Theorem 3. Theorem 2, which is used in the proof of Theorem 3, is a generalization of a theorem recently proved by Loveland (4).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1001 - 1003
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bing, R. H., A surface is tame if its complement is 1-ULC, Trans. Amer. Math. Soc. 101 (1961), 294–305 Google Scholar
2. Burgess, C. E., Characterizations of tame surfaces in E3, Trans. Amer. Math. Soc. 114 (1965), 80–97.Google Scholar
3. Loveland, L. D., Tame surfaces and tame subsets of spheres in E3, Trans. Amer. Math. Soc. 123 (1966), 355–368.Google Scholar
4. Loveland, L. D., Piercing locally spherical spheres with tame arcs (to appear in Illinois J. Math.).Google Scholar