A proof of the generalized Schoenflies theorem

The following problem has been of interest for some time: Suppose h is a homeomorphic embedding of Sn~~1X 0l in Sn. Are the closures of the complementary domains of h (Sn~1Xl/2) topological w-cells? Recently, Barry Mazur l proved that the answer is affirmative if the embedding h satisfies a simple niceness condition. In this paper we prove that the answer is affirmative with no extra conditions on h required. DEFINITIONS AND NOTATION. (1) If Q is an n-cell then Q and Q respectively denote the boundary and interior of Q. (2) / denotes the unit interval 01. (3) If/: X—Fis a map, then an inverse set (under/) is a set M (ZX containing at least two points, and such that for some point y of f (X), M=tl (y) - (4) A set M is cellular in an n-dimensional compact metric space S if there exist /z-cells Qi, Qi, • • • in 5 such that Qi+iQQu and THEOREM 0. Let Q be an n-cell and let f map Q into the n-sphere Sn. Suppose also1 that ƒ has only a finite number of inverse sets, and that these inverse sets are all in Q. Then f (Q) is the union off (Q) and one of its complementary domains. PROOF. Let h=f \\ Q. If ƒ ( (? ) Cf (Q) then h~y maps Q into Q and is fixed on Q. This is impossible, hence ƒ (Q) intersects one of the comple-mentary domains, say D, of ƒ ( () ). Now Q does not separate Q, and ƒ ( (? ) separates Sn; hence f (Q) CD. If f (Q) does not contain S then ƒ ( (? ) has infinitely many boundary points in Z. But by Brouwers theorem on the invariance of domain, and the hypothesis, only a finite number of points oîf (Q) r\ are boundary points of ƒ ( (? ). Hence / (G) «5. THEOREM 1. Let Q be an n-cell. Suppose M is a cellular subset of Q.