Remarks on a Point Set Constructed by Antoine.

which completes the induction for this case. In the remaining case where the point Q is on the curve ko, the only difference is that an arc, 11, of a curve of intersection in a,, and not necessarily an entire curve, approaches the curve ko as a, approaches aio. The necessary deformation of a, is one such that the arc (or curve) 11 shrinks to the point Q as a, approaches ao. We perform a similarly modified deformation on 02 and complete the argument just as before, thereby proving the theorem. A similar reduction may be applied for the case p = 1, but at some stage of the process the curve ko will be non-bounding. The side of a containing the plane surface C bounded by ko will thus have to be tubular, that is to say, homeomorphic with the interior of an anchor ring. This is the theorem predicted by Tietze. For a general value of p, it is easy to show that the linear connectivity of either region bounded by a is (P11) = P, but the group of the region may be very complicated.