( 147 ) 



and we draw a path tv\ joining the infinite with a point Q\ of x,, 

 and abroad from Q^ lying entire!}^ in the outer region of C as well 



as in the onter region of y.^. Then id\ forms together with some 

 parts of u\ and of the radius Qi Qi of Tt^ a path l^ joining M witli 

 the infinite, and abroad from M not meeting C, -j- f, -|- f^. 



In the same way we can construct a path 4 joining Al witli the 

 inlinite, coinciding for a certain initial part with a part of v\, and 

 abroad from M not meeting C^ -\- f ^ -{- f^. 



As in the vicinity of M the arcs f, and f, are separated by w^ 

 and u\, in the complete plane t\ and e^ are separated by /j -)- /^ 

 (whether l^ and 4 meet each other abroad from M or not). 



So, since /, -f- 4 contains no point of C-^^, also />i and p.^ arc 

 separated in the complete plane by l^ -\- /.,. Hence i\ and y?, cannot 

 be identical, and Cj, cannot be a continuum. 



As furthermore two finite continua whose common points form a 

 non-coherent set determine more than one region in the plane '), 

 from theorem 4 ensues immediately : 



Theorem 5. A continuous one-one image of a closed curve divisible 

 into two proper arcs of curve determines in the plane more than 

 one region. 



1) This may be proved by breaking up the boundary of a region determined 

 by the common points of these continua into two closed sets (\ and Co poi;sebsing 

 a finite distance from each other, and then applying the reasoning of Matliem. 

 Annalen, Vol. 68, p. 430. 



