312 



Let AB and AC be the branches arriving at A in « and B' C' a 

 iinesegment crossing both arches AB and AC. The Jordan theorem 

 for three dimensions shows that a double connection exists between 

 the branches AB and AC by means of two sets of points I and II, 

 having no points in common. If I and II were situated on the same 

 side of (( tiien a parallel Iinesegment converging from that side 

 towards B' C' would end up b}- having at least two points in 

 common with I and also two with II: an impossibility. Hence I 

 and II lie on different sides of u and inside anv neighbourhood of 

 A points of i^' exist on both sides of a. 



The vicinity of A on F' is the (1,1) continuous representation 

 of the vicinity of a point in a plane. From this follows that 

 inside any finite neighbourhood of A the points of /?, which are not 

 situated on F^, belong to either of two regions 6-\ and G^, which 

 regions are not connected within that finite neighbourhood of ^. The 

 common boundary of these regions F^ has Jordan character. Inside 

 any vicinity of A we found points of F^ on both sides of <(. 

 Let ylj, A^ . . . be a sequence of these points converging from above 

 and A^', AJ . . . a sequence converging from below towards A. 

 A^ and A^' can be joined by a single path, belonging entirely to 6^1 

 and by another belonging to 6r,. In the same way A^ and A^' etc. 

 By going far enough in the sequence it follows from the Unbewalltheit 

 that these paths can be kept inside an arbitrarily small neighbourhood 

 of A. Each path connects a point above a with a point below a, 

 hence in a points of both G^ and G, exist inside any vicinity 

 of A. From this follows that in tt tlie region on one side of the 

 convex arch BAC belongs to G^ and that on the other side to G,. 



Let D and F be points on line n on different sides of ^. D 

 belongs to G^, E to G,. Round D a finite sphere b^ exists, all 

 internal points of which belong to 6^1 and round F a sphere b^ 

 containing points of G^ only. 



Now^ let the linesequence üi, a, . . . converge towards a. Let on 

 these lines the points D^, D^ . . . converge towards Z) and E^, F, . . . 

 towards F. D^ D^. . . end up by being internal to sphere èj and 

 then certainly belong to Gy. F^ F^. . . become internal to b^ and 

 then belong to G^. Hence for n larger than some finite number the 

 finite Iinesegment Z)„ F„ of a„ carries at least one point of F" and 

 these points can only converge towards A, because this is the only 

 point of i^' on the segment DF of a. 



Theorem 2 : If the planes «,, «, . . . convertje towards a, theti 

 the section of F" in « consists of the limiting set of the sections in 



