1193 



and w the set of intervals wliich is left from f after destroying 

 all points of v\^i contained in / . Let PQ be an arbitrary element 

 of IV, Sp the set of intervals determined as the intersection of Up 

 and FQ, tp the set of intervals which is left from Sp after 

 destroying its first and its last element, in so Jar those elements 

 exist, y the set of intervals determined as the union of ^i, t^, t„ . . . , 

 and (p the set of intervals which is generated by constructing 

 in each element of lu a set of intervals in the same way as y has 

 been constructed in PQ. Then the required set of intervals /v-j-i is 

 generated by constructing in each element of ;'„ a set of intervals 

 in the same way as rp has been constructed in AB. 



If we understand by u^ as well as by j„ the unit interval itself, 

 then we arrive from j^ at ƒ, by the same process which has led us 

 from j\ to jv_|-i. 



The inner limiting set determined as the intersection of Ji, J,, . . . 

 contains all points of U belonging to none of the sets v-,, so a 

 fortiori all points of U belonging to none of the sets a,,, so also 

 all points of the unit interval belonging to none of the sets a^. As, 

 on the other hand, this inner limiting set can neither contain a 

 point of a v^, nor (as a subset of U) a point of a j)^, it finally 

 cannot contain a point of a a^ either. So it is identical to the com- 

 plementari/ set of C, i.e. to I. 



If we construct a ternal scale on the unit interval, and if (designing 

 by Qv an arbitrary finite series of digits 0, 1 or 2, among which 

 V digits 1 occur) we understand by f/v-fi the set of the intervals 

 Ip whose end-points have the coordinates -^vl and -^„2, then we can 

 first represent the set of intervals j\, biuniformly and with invariant 

 relations of order, on the set of intervals ^i ; thereupon we can in each 

 interval of j^ represent the subset of j", contained in it, biuniformly 

 and with invariant relations of order, on the subset of d, contained 

 in the corresponding interval of d^ ; and so on. In this way we 

 determine a continuous one-one transformation of the unit interval 

 in itself by which I passes into the set t, of the points expressible 

 in the ternal scale by means of a sequence of digits containing an 

 infinite number of digits 1, whilst C passes into the set t, of the 

 points expressible in the ternal scale by means of a sequence of 

 digits containing only a finite number of digits 1. Thus, indicating 

 the geometric types'^) of t, and t, by n and v respectively, we have 

 proved the following 



Theorem 1. Eaclt inner limiting set contained in a linear interval, 



1) Gorap. these Proceedings XV, p. 1262. 



