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open Jordan curves A^R^ and A^S^ in three regions having no points 

 in common. In the vicinity of A^ all these regions have the character 



Ffg. 3. 



of Jordan regions. We consider the two outside regions, namely 

 those connecting respectively A^E^ with A^R^SiUO A\S^ with A^D^^). 

 The (1,1) continuous representations of these regions of F^ connect 

 respectively A E with AR and .1-S with AD. That this connection 

 exist§ inside any neighbourhood of A, again follows from tiie "Un- 

 bewalltheit". Hence in any plane through b (tig. 3) such that AE 

 and AR are situated on different sides, at least two branches arrive 

 at A from below a. But we also know that in each of these planes 

 two branches arrive in A from above a (one on 1 and the other 

 on III), hence the following result has been obtained : When the 

 plane [3 is turned round b (fig. 3) in such a way that the lower half 

 moves to the left, then in every position as far as « the point A 

 remains double point. 



Let c be a line in « through A, passing between the branches ^£' 

 and AD, and let d be a line in [3 through .4, separating the branch- 

 es AR and AS. The plane through c and d is denoted by y (fig. 3). 

 In y two branches arrive in A from below «, one on II and the 

 other on IV. The branch situated on II arrives' in A from the right 



1) A priori it would be possible that ^i-^i is connected with AiSi and A^Ri 

 with AiDi, but when we consider the representations on F'^, this leads to contra- 

 diction with the Jordan theorem for threedimensional space. 



