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A P R O O F O F T H E G E N E R A L IZ E D S C H O E N F L I E ST H E O R E M

BY MORTON BROWNCommunicated by Edwin Moise, January 4, 1960

The fol lowing problem has been of interes t for some t ime : Supposeh i s a homeomorphic embedding of Sn~~1X [0l ] in Sn. Are the c losuresof the complementary domains of h(S n~1Xl/2) topological w-cells?Re cent ly , B ar ry M azu r [ l ] p roved th a t the answer is a f fi rmat ive ift h e e m b e d d i n g h sat isf ies a s imple "niceness" condi t ion. In this paperwe pro ve th a t the answer is af f irmat ive w i th no ex tra co ndi t ion s onh r equ i r ed .D E F I N I T I O N S A N D N O T A T I O N .(1) If Q is an n-cell t h e n Q an d Q r e spect ive l y deno t e t he b ou nda ryand inter ior of Q.(2) / d eno tes th e uni t interv al [01 ] .(3) I f / : X >Fis a m ap , the n an inverse set (u nd e r / ) i s a s e t M(ZXconta in ing a t l eas t two poin t s , and such tha t for some poin t y off(X), M=tl(y)-(4) A set M is cellular in an n-d imens iona l compact met r ic space

S if there exist /z-cells Qi, Qi, in 5 such th a t Qi+iQQu an dT H E O R E M 0. Let Q be an n-cell and let f map Q into the n-sphere Sn.

Suppose also1 that has only a finite number of inverse sets, and thatthese inverse sets are all in Q. Then f (Q) is the union off(Q) and one ofits comp lementary dom ains.

P R O O F . L e t h=f\ Q. If ((?) Cf(Q) t h e n h~y m a p s Q in to Q and isfixed on Q. T hi s i s impossible , hen ce (Q ) intersects one of the complem e n t a r y d o m a i n s , s a y D, of (()) . Now Q does no t separa te Q, and((?) separates Sn; hence f(Q)CD. If f(Q) does no t conta in S then((?) has inf inite ly m an y bo un da ry po ints in Z>. B u t by Br ou w er 'stheorem on the invar iance of domain , and the hypothes i s , on ly af ini te number of points of(Q)r\D are boundary poin t s of ((?). H e n c e/(G) 5 .

T H E O R E M 1. Let Q be an n -cell. Suppo se M is a cellular subset of Q.1 Actually the theorem is true if we remove the finiteness condition. Then onecould prove the theorem by methods such as those in Hurewicz and Wallman's book(page 97).

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A PROOF OF TH E GENERALIZED SCHOENFLIES THEOREM 75

Then there is a map of Q onto itself such that is fixed on Q and M isthe only inverse set under f.

P R O O F . L e t Qi be a sequence of n-cells in Q whose in tersec t ion i sM a n d s u c h t h a t Qi+i C Qi> L e t hi be a homeomorph i sm of Q onto i t self which is fixed on Q and such tha t the d iameter of h(Qi) is lessth an 1 . Ind uc t ive ly , le t hi+i be a homeomorphism of Q upon itselfs u c h t h a t hi+\hi on Q Qi and such tha t the d iameter of h i+ i(Q i+ i)is less th an l / ( i + l ) . S ince ||A*- A*+I|| i s less than l / ( i + l ) , / = lim t- hiis a map of Q u p o n itself. T h e hi are all fixed on Q, he nc e s o is . Ifxy y are two different points of Q not both in M, then one of them,sa y xy does not lie in some cell Qi. Then/(x) = &(#). I f y is in M t h e nf(y) is in hi{Q x). But /^-(x) does not lie in hi{Q %). Hence if y is in Mthen (x) 5ef (y). If ^ is not in M then for some j , ne i the r x nor y liesin y. But then f(x)=hj(x)y and f (y) = hj(y). B u t ^ is a homeo mo rp h i s m , so f(x)5*f(y). Fin al ly , i t is clear from th e con stru ct io n of t h a t f(M) i s a point . Hence M i s t he on ly inve r se se t under / .

T H E O R E M 2 . Let S be a topological n \ sphere in S% and let D beone of its complementary domains. Suppose f map s D onto an n-cell Rsuch that the only inverse set of f is a cellular subset M of D. Then Dis an n-cell.

P R O O F . L e t Q be an n-cell in D s u c h t h a t MCQ- T h e n M is cellularin Q. H e nce by Th eorem 1 the re i s a map g of Q onto i tself such thatg is fixed on Q and the only inverse se t under g is M. L e t g' be the mapof D onto i tself which is the identi ty on D Q a n d g on Q. T h e n t h em a p fg'~ of D o n t o R i s a homeomorph i sm. H ence D is an n-cell .T H E O R E M 3 . Let Q be an n-cell and suppose f maps Q into S n. Sup

pose also that MCQis the only inverse set under f. Then M is cellularin Q.

P R O O F . I t fo llows f rom Th eor em 0 th a t (Q) = ( ( ) )U D whe re Di s a com ple m en tary do m ain of ( ( ) ) . Le t U be an open subset of Qc o n t a i n i n g M. T h e n \U ) is open in D and con ta ins (M) . L e t h bea homeomorph i sm of 5W upon i tself which carr ies D in to ( / ) andwh ich is f ixed on some (sm all) neigh bor hoo d F of the poin t (M). L e tg be the map of Q into itself defined by:

(x, xE M yg(x) =