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Kirby has given an elementary geometric proof showing that if an (n — l)-sphere £ in Euclidean n-space E is locally flat modulo a Cantor set that is tame relative to both 2 and e", then X is locally flat. In this paper we illustrate the sharpness of the result by describing a wild (n l)-sphere S in e" such that E is locally flat modulo a Cantor set C and C is tame relative to E . These examples then are used to contrast certain properties of embedded 3 spheres in higher dimensions with related properties of spheres in E . Rather obviously, as Kirby points out in [11], his result cannot be weakened by dismissing the restriction that the Cantor set be tame relative to En. It is well known that a sphere in E containing a wild (relative to E ) Cantor set must be wild. Consequently the only variation on his work that merits consideration is the one mentioned above. The phenomenon we intend to describe also occurs in 3-space. Alexander's horned sphere [1] is wild but is locally flat modulo a tame Cantor set. In fact, at one spot methods used here parallel those used to construct that example. However, other properties of 3-space are strikingly dissimilar to what can be derived from the higher dimensional examples constructed here, for, as discussed in §2, natural analogues to some important results concerning locally flat embeddings in E are false. Most of the terminology and notation is standard. We distinguish between the two senses of the term "boundary" by using 971/ to denote the boundary of a manifold M and Bd A, for A C X, to denote the boundary of A in the space X. Our standard &-cell Bk is the set of points in Ek of norm < 1. We use p to denote the standard complete metric on E"+', and for two maps/and /' of a space X into E"+!, we use p(f, /') to denote lub {p(f(x)>f(x))\x G X}. 1. Construction of certain wild spheres. The somewhat intricate definition and lemma that follow are designed to slip naturally into the proof of Theorem 3 Received by the editors February 2, 1973 and, in revised form, February 22, 1974. AMS (MOS) subject classifications (1970). Primary 57A15, 57A45, 57A50; Secondary 57A60.

Wild spheres in $E\sp{n}$ that are locally flat modulo tame Cantor sets