Vol. 19, No. 3, 1966

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Tame subsets of spheres in E3

Lowell Duane Loveland

Vol. 19 (1966), No. 3, 489–517

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 Smisses F (that is, S Slies 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,,Fn is a finite collection of closed subsets of S such that Property (,Fi,S) holds for each i, then Property (, Fi,S) also holds. We use this result to show that if S is locally tame modulo Fi, then S is tame.

Mathematical Subject Classification
Primary: 54.78
Received: 26 July 1965
Published: 1 December 1966
Lowell Duane Loveland