Sufficient conditions for a closed set to lie on the boundary of a 3-cell
L. D. Loveland · Proceedings of the American Mathematical Society · 1968
Let F be a closed subset of a 2-sphere S in S3, and let V be a conmponent of S3-S. Using some results and proofs from [7] and [8] we give several conditions that are sufficient for F to lie on the boundary of a 3-cell. One such condition is Property (*, F, V) which is defined in [7]. Property (*, F, V) roughly means that the polyhedral approxiination S' to S obtained using Bing's Side Approximation Theorem [2] can be chosen such that S' lies almost in V and SnS' lies in the union of a finite collection of disjoint small disks in S-F. Property (*, F, S) is defined in [7] to mean that Property (*, F, V) is satisfied for each component V of S3-S. The author [7] established a conjecture made by Gillman [5] and then used some of Bing's techniques [1] to show that F lies on a tame 2-sphere if Property (*, F, S) is satisfied. Unfortunately the following theorem was not included in [7] and is not a direct consequence of any of the results there.