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8 MA THEMA TICS: J W. ALEXANDER PROC. N. A. S. 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 (P1 - 1) = P, but the group of the region may be very complicated. AN EXAMPLE OF A SIMPLY CONNECTED SURFACE BOUNDING A REGION WHICH IS NOT SIMPLY CONNECTED BY J. W. ALZXANDER DIPARTMENT OF MATHUMATICS, PERNCETON UNIVERSITY Communicated, November 19, 1923 The following construction leads to a simplified example of a surface 2 of genus zero situated in spherical 3-space and such that its exterior is not a simply connected region. The surface 2 is obtained directly without the help of Antoine's inner limiting set. The surface 2 will be the combination, modulo 2, of a denumerable infinity of simply connected surfaces Si (i = 1, 2, . . . ), all precisely similar in shape, though their dimensions diminish to zero as i increases without bound. The shape of the surface Si may perhaps be described most readily by referring to the accompanying figure in which the surfaces 52 and S3 are represented. By