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 Brouwer's 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.