( 144 ) 



possesses not only each special numerical value but also the value 

 of each dual fraction not identical to or 1 over a certain interval 

 of abscissae, then the just now described construction of the gene- 

 ralized figure 1 can be repeated without modification with the only 

 difference that the supporting threads are replaced by supporting 

 bands. These supporting bands determine together with the rest region 

 of F a region G possessing the same boundary g as the inner 

 domains of the coherence bands. 



So this continuum g divides the plane into regions with a common 

 houndai-y ; whether the number of these regions is finite or infinite, 

 depends on tlie choice of the special directly coherent sets of numbers. 



Let us call two points contained in a perfect coherent part of a 

 not identical to g, directhj coherent in g, and let us call the set 

 formed by tlie points directly coherent in g with a given point, a 

 72erve of q, then the skeleton of the figure and likewise each cohe- 

 rence thread furnishes one nerve of g, and each coherence band 

 furnishes two nerves of g. 



If we choose oidy one special directly coherent set of numbers, 

 then our construction furnishes a closed curve (in the sense of 

 ScHOENFLiEs) which Can be divided into two improper arcs of curve 

 but not into two proper ones, in which category is included the 

 primitive figure from which we started. 



M- 



The hnposslblUtg of a linear arrangeinent of the points of an 

 irreductible continuum. 



By ZoRETTi lately a method has been explained of arranging the 

 points of an irreductible continuum linearly, analogously to those 

 of a line segment^). 



His method is however inapplicable to several continiia constructed 

 in the article ''Ziir Analysis Situs" cited above. 



This having been pointed out to him, Zoretti has based a method 

 of more restricted aim on the following theorem ') : 



"Given an irreductible continuum C and a point c of C, then C 

 can be divided in one depnitr. manner into three sets of points C, , C', 

 and r, possessing the follow ring properties : C^ and Q are coherent 

 and have c as their only common point; r consists of the common 

 limiting points of C, and €,. Both sets of points C^ + FandC^ + r 

 are irreductible continua." 



1) Annates de TÉcole Normale, 1909, p. 485—497. 



2) Comptes Rendus, t. 151, p. 202. 



