110 



finish up bj having at least one point in common with each of II 

 and IV, and at least two with I, and this again cannot be as no 

 line carries four points of F^. Hence the final result is that I and 

 III are situated above « and II and IV below u or vice versa. 



A representative case of the second possibility is the following: 

 AC and AE are connected by I above a, AE and AF above or 

 below a by II, AF and AD below « by III and lastly ^Z) and ylC 

 above or below a by IV. If IV be situated below « we choose in 

 a a point A' near A and a point D' near Z), such that the line- 

 segment A' D' intersects the arch AD at a point near A and at 

 another point near D. Now a parallel linesegment converging from 

 below towards A' D' would end up by carrying at least two points 

 of III and two of IV: a contradiction ^). Hence the second possibility 

 left by the Jordan theorem is excluded and we need only consider 

 the first. In the following it will be assumed that I and III are 

 situated above, and II and IV below «. 



Obviously the set of points \ -\- AC -\- AD is the (1,1) continuous 

 representation of a plane region and part of its boundary. Besides, 

 inside a finite neigh bom-hood of the point corresponding to A, this 

 region has the character of a Jordan region, because the arches 

 AC and AD are Jordan curves, and the same holds for the (1,1) 

 continuous representations. The same things can be said oi\\-\-AD \-AE, 

 III 4_ AE-i-AF and lY -^ AF -[- AC 



Lastly we remark that inside a finite neighbourhood of ^ all points 

 of F\ not situated in « belong to I -[- II + III -f IV. 



Let 6 be a line in a through A such that the branches FA and 

 EA arrive at A from different sides of this line. Then the branches 

 CA and DA will do the same. Let /? be a plane through h {=\— a). 

 AC and AD are joined above u by I. i -\- AC -\- AD is the 

 continuous (1,1) representation of a plane region and part of its 

 boundary. Let I^ correspond to I, A^C^ to AC, emid A^D^ to AD. 

 Inside a finite neighbourhood of A^ the region I, has the character 

 of a Jordan region. 



We shall now have to use a property of Jordan regions called the 

 "Unbewalltheit". ^) For two dimensions it may be formulated as 

 follows: Let i/ be a closed Jordan curve, I the internal and E the 

 external region. Two points Q and R of J can always be joined 

 by an open Jordan curve belonging entirely to 7 and by an open 



1) By using this last reasoning the first possibility might have been dealt with 

 in a more simple fashion. 



2) Brouwer, Math. Ann. 71, p. 321. . 

 ScHOENFLiES, Mengenlehre 2, chapter 5. 



