Almost locally polyhedral curves in Euclidean 𝑛-space
James C. Cantrell, C. H. Edwards · Transactions of the American Mathematical Society · 1963
Fox and Artin [2] have given examples of wild arcs and curves in £3 which fail to be locally polyhedral at only one or two points.It is shown in this paper that no such "simple" 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 Euclidean «-space E", « > 3, must fail to be locally polyhedral at each point of a Cantor set.Examples of such wild curves in £" have been given by Blankinship [1].A set K in E" is called tame if there is a homeomorphism h of E" onto itself such that h(K) is polyhedral (relative to the standard triangulation of £").Otherwise K is wild.K is said to be locally polyhedral at the point peK if there exists a neighborhood N of p such that Cl(N C\K) is a polyhedron.The map «:£"->£" is said to be locally semilinear at x if there is a neighborhood N of x such that « | N is semilinear.In this paper, S(p,e) denotes the set of points xe£" whose distance p(x,p) fromp is less than e.The local connectivity of an arc gives the following lemma.Lemma 1. Suppose that A is an arc in E" with p an interior point of A. Given e > 0, there exists ô >0 such that, if L is any subarc of A whose endpoints lie in S(p,ö), then L c S(p,s).Lemma 2(2).Suppose that C is a simple closed curve in E", « > 3, and that B is the set of points at which C fails to be locally polyhedral.If p is an isolated point of B, then, given e > 0, there exists a homeomorphism h of E"onto E" such