A Note about Locally Spherical Spheres
W. T. Eaton · Canadian Journal of Mathematics · 1969
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).