( 145 ) 



From this theorem would follow that, if not all the points of C, yet 

 a considerable part of them would be capable of linear an-angement, 

 and that it would be possible to crumble 6' in the same way as a 

 line segment into an indefinitely large number of lineai-ly arranged 

 ''partial arcs", two arbitrary ones of which then and onl}- I hen 

 cohere, if in that linear order they succeed each other immediately. 



But neither this theorem can be maintained, if we try to apply 

 it to our primitive figure. 



If namely we choose this closed curve as the irreductible continuum 

 C, then either Ci -f- -T or C, -\- r must be identical to C, and either 

 Cj or Cj reduces to the single point c, so that the division of C 

 becomes illusory. 



It is a priori certain that all attempts to arrange the points of 

 such a continuum linearly by repeated crumblings must fail, the 

 crumbling being practicable only for a single system of directly 

 coherent points, and therefoi'e the linear arrangement being restricted 

 in any case to points of a single nerve. 



And even of this we are not sure for the most general irreductible 

 continuum. For, in a system of points directly coherent in C again 

 may be contained an irreductible continuum C' breaking up into 

 a set of the power of infinity of the continuum of systems of points 

 directly coherent in C' . And so on. 



§ 5. 



A generalization of Jordan's theorem. 



Jordan's theorem runs that a continuous one-one image of a circle 

 is a closed curve, i.e. divides the plane into two regions of which 

 it is the common boundary. 



The extension lying at hand that a continuous one-one image of 

 a closed curve is again a closed curve, has not yet been proved. 

 However, for a special kind of closed curves a partial result can 

 be arrived at, as we shall explain in the following. 



Let C be an arbitrary closed curve, and let us represent by 

 C"»i/ the cyclic type of order of its points accessible from its iiuicr 

 region. A Schnitt s arbitrarily given in c'*ii determines two "Schnitt- 

 contijiua" oi and (J,., to which cVi converges on the left resp. on 

 the right of s. The points common to <J/ and (7,. form a closed set of 

 points o, to be called "the juncture helongimj to the Schnitt s". 



Lemma. In the inner region of C we can construct an arc oj 

 curve which abroad from its ends is simple, and of lohich one end 

 reduces to a single point of the inner region of C and the other is 

 contained in the juncture o. 



