( 139 ) 



sets of pieces possess tlio same ficoinelric typo ol' ()r<K>r, namely llio 

 common type ol* onier of linear, perfect, piincliial sets of points. An 

 analogous theorem exists for closed sets of |)ieces. 



In § 3 of the first comnnmication the set of pieces n, taken there 

 as perfect, was broken np into two such closed sets fj,, and </„, that. 

 (Hull) "^ f Hi') iiii<l "(i'^) f'i) ^ '^Mi'O? llicn each m, into two such closed 

 sets ƒ!/,„ and ;t/,2, that (f{ni,L)^(Hnii) 'Mk\ (tin ha, !.ii,i)> (i{(.ih); auó so on. 

 In this way the f/, 's converged for indetinite accresccncc of the rows 



of indices / unifoi*ndy to the pieces of (i, and we could construct a 

 continuous one-one correspondence hotweeu the pieces of (i and a, 

 nowhere dense perfect set of real uund)ers hetween and J, where 

 for each of those nninl)ers the row of tigures in the numei-ical system 

 of base 3 was identical to the row of indices of the corresponding 

 piece of ft. 



If, however, ;< is a closed, not |KM-fect set of pieces, then the 

 breaking uj) of an arbitrai'y jr^ into ƒ*, ,) and (if .^ can take j)lace in 



the same way with the only exception that a u^ consisting of a 



single piece also appears as (i^ „ , whilst ft^ ,, falls out. Then too the 



fi, 's converge for indetinite accrescence of the rows of indices / 



uniformly to the pieces of (i, and we can construct a continuous 

 one-one corresj)ondence between the [)ieces of n and a nowhere 

 dense closed set of real numbers between aiul 1, where for each 

 of those numbers the I'ow of tigures in the numerical system of 

 base 3 is identical to the row of indices of tiie corresponding 

 piece of [i. 



So we have proved : 



Theo]{EM 3. Each closed set of pieces in S/),, possesses the (/eotnetric 

 type of order of a linear, closed, punctual set of points. 



§ 3. 



I'/te (linision, of the phrne into more than tino regions 

 uritli a common houndarij. 



On a former occasion ') I constructed a division of the plane 

 into three regions with a common boundary, and 1 communicated 

 at the same time that by a suitable modification of the method 

 followed there a division into an arbitrary (inite number, and even 

 into an infinite number of regions with a common l)oundary can be 

 obtained. That modified method I shall now explain. 



1) Compare "Ziir Analysis Situs", Matiiem. Annalen, Vol. 08, p. 422—434. 



