Tame subsets of spheres in E³

Lowell Duane Loveland

Vol. 19 (1966), No. 3, 489–517

DOI: 10.2140/pjm.1966.19.489

Abstract

Let F be a closed subset of a 2-sphere S in E³. We define F to be tame if F lies on some tame 2-sphere in E³. The sets F and S satisfy Property (∗,F,S) provided Bing’s Side Approximation Theorem can be applied in such a way that the approximating 2-sphere S′ misses F (that is, S ∩ S′ lies in a finite collection of disjoint small disks in S − F). In this paper we show that Property (∗,F,S) implies that F is tame by establishing a conjecture made by Gillman. Other properties which are equivalent to Property (∗,F,S) are also given.

If F₁,F₂,⋯,F_n is a finite collection of closed subsets of S such that Property (∗,F_i,S) holds for each i, then Property (∗,∑ F_i,S) also holds. We use this result to show that if S is locally tame modulo ∑ F_i, then S is tame.

Mathematical Subject Classification

Primary: 54.78

Milestones

Received: 26 July 1965

Published: 1 December 1966