1194 



and, as well as its complementary set, uncountable in each sab-interval, 

 possesses the geometric ti/pe i', and its complementary set possesses 



the geometric type fi. 



Let H and K be two arbitrary points of t,, we can choose v in 

 such a way that neither H nor li is an endpoint of an interval of 

 d,. Let us indicate the set of points whicli is the complementary 

 set of d^, by e-,, and (he set whose elements are the intervals ci d^, 

 and the points of e.„ by r,. Then we can construct a one-one tians- 

 formation of r/, and ^v each in itself, by which the relations of order 

 between the elements of 7', remain invariant, and H passes into A. 

 This transformation can be extended to a continuous one-one tra s- 

 formation of the unit intei-val in itself, for which the subsets of r^ 

 and T, contained in corresponding intervals of r/y, correspond to each 

 other. We thus have generated a continnous one-one transformation 

 of tlie unit interval in itself, by which Tj passes into itself, and the 

 point H chosen arbitrarily in t,, passes into the point K chosen 

 likewise arbitrarily in t,, so that t, is a homogeneous set of points. 



Let H and K be two arbitrary points of r, contained in the 

 intervals Av and k^ of d, respectively, we can construct a one-one 

 transformation of the set of intervals d^ in itself leaving invariant the 

 relations of order, by which A, passes into k^ ; this transformation 

 can -be extended to a one-one transformation of the set of intervals 

 d^ in itself leaving invariant the relations of order, by which /i, 

 passes into k^\ continuing indefinitely in this way, we generate a 

 coniinuons one-one transformation of the nnit interval in itself, by 

 which each d,, so also t^ passes into itself, and the point //chosen 

 arbitrarily in t,, passes into the point K chosen likewise arbitrarily 

 in T^, so that r^ too is a. homogeneous set of points, and we have 

 proved the following 



Theorem 2. Each inner limiting set contained in a linear interval, 

 and, as ivell as its complementair y set, uncountable in each sub-interval, 

 is homogeneous, and its complementary set is likewise homogeneous. 



