104 



In a plane in which A is isolated the remaining points belonging 

 to F^ form a connected curve of the third order or a straight line. 

 This restcurve is a closed set of points (it is the continuous repre- 

 sentation of a circle), hence A has a finite minimum distance from 

 it. Let this minimum distance be 8^ in i^^, e^ in i?, etc. When a point 

 B moves along the restcurve in (?, the distance AB changes 

 continuously from f, to oo, in j^, from 8, to oo etc. (when a point 

 A is situated at distances b^ and b, from two points B^ and B^ 

 belonging to a connected set of points then to every distance b^ such 

 that />! > 63 ^ A3 corresponds at least one point B^ of the set such 

 that AB, = b,). 



The sequence e^, e^, e^ . . . . has zero for limit. Let ^i, rf^, d, . . . . be 

 a decreasing sequence chosen from it and let the corresponding 

 planes be represented by |?,;,, ^c.,, i^^^ .... 



In jj,;^ we choose a point B^' of F^ such that rfj ]> AB^' ^ ö^ 



and „ „ ^3' ,, „ ,, ,, 6^ > AB^ > (f, 



In jio^ we choose a point B^" „ ,, ,, ,, (f^"^ AB^" '^ (f, 



and „ „ i?," „ „ „ „ <f, > ^^3" > d, 



and „ „ .0/ „ „ „ „ dj > AB^ > (f, 



etc. 



i?2', B^', BJ". . . . have a limiting point B^ in ^ such tiiat dj ^ x^l^, ^ tf, 



i^;, B,", B:\ . . . „ „ „ „ B, „ ^ „ „ d, ^ ^^3 ^ tf, 



etc. 



/^' is a closed set of points, hence B^, B^, B^. . . all belong to i''\ 



Besides d^, d^, d, is a decreasing sequence having zero for limit 



hence A is limiting point of F^ in /l 



We now proceed to construct a finite sphere round A inside of 

 which F^ is entirely situated on one side of the plane a (except the 

 point A in a). A is isolated in rf, hence with A as centre there 

 exists in a a finite circle c containing no other points of F^. Let 

 b be the sphere with A as centre passing through c. The vicinity 

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

 of a point in a plane. Let A^ be the point corresponding to A. 

 The correspondence is (1,1) continuous, hence a finite circle c^ 

 round A^ can be found in the plane such that all internal points of 

 6*1 have corresponding points inside the sphere b and a sphere b' con- 

 centric with b can be found such that all internal points of b' 

 belonging to F^ have corresponding points inside Cj. 



Inside b' F^ lies on only one side of a for if this were not the 

 case, a contradiction might be obtained as follows: Two points B 



