( 788 ) 
To each destruction of an isolated point or isolated piece in p now 
answers a destruction of at least one 4 ) cube in K x ; but of the latter 
destructions only a denumerable number is possible, thus also of the 
former, with which Cantor’s theorem and Schoknflies’s theorem are 
proved both together. 
Let us call the rest set, which remains after destruction of all 
isolable pieces, the Sehoenflies residue, then on the ground of theorem 
1 we can formulate: 
Theorem 2. A Sehoenflies residue is a perfect set of pieces. 
§ 3. 
The structure of perfect sets of pieces. 
Let S t and S 3 be two pieces of a perfect set of pieces p. Let it 
be possible to place a finite number of pieces of p into a row having 
St as its first element and S, as its last element in such a way, that 
the distance between two consecutive pieces of that row is smaller 
than a. Then we saj, that S, belongs to the a-group of S x . 
If S 2 and S 3 both belong to the a-group of S l} then S s belongs 
also to the a-group of S 2 , so that p breaks up into a certain number 
of “a-groups”. This number is finite, because the distance of two 
different a-groups cannot be smaller than a. 
If a x a,, and if an a x -group and an a,-group of p are given, 
then these are either entirely separated or the a x -group is contained 
in the a 9 -group. 
If two pieces S t and S 2 of p are given, then there is a certain 
maximum value of a, for which S x and S 2 lie in different a-groups 
of p. That value we shall call the separating bound of S x and S, in 
p, and we shall represent it by (S x , S 2 ). 
If fartheron we represent the distance of S x and S z by a (S x , S 2 ), 
then (St , S 2 ) converges with a (S x , S 2 ) to zero, but also inversely 
« (S t , S 2 ) with <j m (S x , S 2 ). For otherwise convergency of a /x (S x , S 2 ) to 
zero would involve the existence of a coherent part of p, in which 
two different pieces of p were contained, which is impossible. 
The maximum value of a for which p breaks up into different 
a-groups we shall call the width of dispersion of p, and shall represent 
it by <f (p). This width of dispersion of p is at the same time the 
greatest value which <?>(£*,&,) can reach for two pieces S x and S t 
of p. 
*) Even of an infinit e number. 
