A Note about Locally Spherical Spheres

Eaton, W. T.

Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1001-1003

Résumé

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).

Détail

DOI : 10.4153/CJM-1969-110-8

Bibliographie

[1] 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] 2. Burgess, C. E., Characterizations of tame surfaces in E3, Trans. Amer. Math. Soc. 114 (1965), 80–97. Google Scholar

[3] 3. Loveland, L. D., Tame surfaces and tame subsets of spheres in E3, Trans. Amer. Math. Soc. 123 (1966), 355–368. Google Scholar

[4] 4. Loveland, L. D., Piercing locally spherical spheres with tame arcs (to appear in Illinois J. Math.). Google Scholar