Vol. 22, No. 3, 1967

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A generalization of the Wilder arcs

P. H. Doyle, III and John Gilbert Hocking

Vol. 22 (1967), No. 3, 397–399

Fox and Harrold first used the words “Wilder arc” to describe a wild arc in euclidean 3-space E8 which is the union of two tame arcs meeting only in a common endpoint and which is locally peripherally unknotted (L.P.U.) at this point of intersection. Thus there are imposed (a) conditions upon the embeddings of subarcs of the wild arc and (b) conditions upon the manner in which the subarcs meet. The following definition gives only conditions of type (a): An arc A E3 is almost tame if each point of A lies on a tame subarc of A. Clearly, every Wilder arc is almost tame.

The chief result characterizes the set W of points on an almost tame arc at which the arc can fail to be locally tame. In particular, W is shown to be homeomorphic to a closed countable set Won the unit interval with the property that a point x Weither is the first or last point of Wor x has either an immediate predecessor or an immediate successor. Two further results discuss special cases.

Mathematical Subject Classification
Primary: 54.78
Received: 6 April 1966
Published: 1 September 1967
P. H. Doyle, III
John Gilbert Hocking