Let K denote the closure of the
interior of a 2-sphere S topologically embedded in Euclidean 3-space E8. If K −S is
an open 3-cell, McMillan has proved that K has at most one nonpiercing point. In
this paper we use a more general condition restricting the complications of K −S to
describe the number of nonpiercing points. The condition is this: for some fixed
integer nK − S is the monotone union of cubes with n holes. Under this hypothesis
we find that K has at most n nonpiercin g points (Theorem 5). In addition,
the complications of K − S are induced just by these nonpiercing points.
Generally, at least two such points are required, for otherwise n = 0 (Theorem
3).
