Almost locally polyhedral curves in Euclidean 𝑛-space

Abstract

Fox and Artin [2] have given examples of wild arcs and curves in E3 which fail to be locally polyhedral at only one or two points. It is shown in this paper that no such examples of wild curves are to be expected in dimensions higher than three. In particular, it is proved that a wild simple closed curve in Euiclidean n-space En, n > 3, must fail to be locally polyhedral at each point of a Canitor set. Examples of such wild curves in En have been given by Blankinship [1]. A set K in En is called tame if there is a homeomorphism h of En onto itself such that h(K) is polyhedral (relative to the standard triangulation of En). Otherwise K is wild. K is said to be locally polyhedral at the point p e K if there exists a neighborhood N of p such that Cl(N n K) is a polyhedron. The map h :En En is said to be locally semilinear at x if there is a neighborhood N of x such that h I N is semilinear. In this paper, S(p, 8) denotes the set of points x e EM whose distance p(x, p) from p is less than B. The local connectivity of an arc gives the following lemma.