1. Introduction. A closed set X in Euclidean 3-space £3 is called tame if there exists a homeomorphism h of E3 onto itself such that h(X) is a polyhedron. A set which is not tame is called wild. In this paper, we investigate conditions which determine tameness of an arc in E3. Examples of wild arcs in E3 are abundant ; see, for example, [3; 8]. Also abundant are conditions implying tameness of an arc; see [7; 10]. Consider the following conditions placed on an arc sé in E3 : (1) sé lies on a

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... phere S in E3. (2) sé lies on a simple closed curve J in E3 which is the intersection of a nested sequence of (two-dimensional) tori plus their interiors. This paper was motivated by a belief that (1) and (2) implied that sé is tame. This turns out not to be the case ; the wild arc constructed in [1] is a counterexample. With this in mind, we make the following definition. A sequence {Ml,M1,---} of 2-manifolds in E3 is sequentially 1-ULC if, given e > 0, there exists a 8 > 0 and integer N such that : Whenever n> N, and a is a simple closed curve on M" of diameter less than Ô which bounds a disk on M", then a bounds a disk of diameter less than e on M". We now add another condition. (3) The sequence of tori of condition 2 is sequentially 1-ULC. Our primary result is that these three conditions imply tameness of the arc sé. This theorem yields as a corollary an answer to a question raised by Bing in [3]: No subarc of the "Bing sling" [3] lies on a disk. A simple closed curve J is said to pierce a disk D if J links Bd D (boundary of D) and J O D is a single point. As the "Bing sling" is the only example in the literature of a simple closed curve that pierces no disk, one is now led to a natural question. Can a different simple closed curve Jf be constructed where Jf pierces no disk, yet lies on a disk? In §3, we show the existence of such a simple closed curve $C. That Jf lies on a disk will be immediate from its construction. To show that Jf pierces no disk, we will use the following. Define P¿ to be the set of points of an arc sé at which sé pierces a disk. We set up an alternate condition to (3) given above. (3') Pj is dense in sé.