( 786 ) 
Theorem 1. Each piece of p is either a limiting piece, or an isolated 
piece. 
Let namely S be a non-isolated piece, then there exists in p a 
fundamental series of points t lt t„t s , ... . not belonging to S, con¬ 
verging to a single point t of S. If t 1 lies on S ly then S x has a 
certain distance e 1 from S. There is then certainly a point f, of the 
fundamental series possessing a distance < s, from S, lying therefore 
not on S x but on an other piece S t . Let t, be the distance of 
from S, then there is certainly a point of the fundamental series 
possessing a distance < e s from S, lying thus neither on S lt nor 
on aS 2 , but on a third piece S t . Continuing in this manner we 
determine a fundamental series of pieces S 1} S t , S t , .. ., containing 
consecutively the points t lf t'„ t\, ... converging to t. So the pieces 
S s , . .. converge to a single limiting piece which can be no 
other than S. 
By a perfect set of pieces we understand a closed set, of which 
each piece is a limiting piece. 
A perfect set of pieces is also perfect as set of points ; but the 
inverse does not hold. For, a perfect set of points can very well 
contain isolated pieces. 
We shall say that two sets of pieces possess the same geometric 
type of order , when they can be brought piece by piece into such 
a one-one correspondence, that to a limiting piece of a fundamental 
series in one set corresponds a limiting piece of the corresponding 
fundamental series in the other set. So in general a closed set 
considered as a set of pieces possesses not the same geometric type 
of order as when considered as a set of points. 
A closed set we shall call punctual , when it does not contain a 
coherent part, in other words when all its pieces are points. 
§ 2 . 
Cantor's fundamental theorem and its extensions. 
The fundamental theorem of the theory of sets of points runs as 
follows: 
If we destroy in a closed set an isolated point, in the rest set 
again an isolated point , and so on transfnitely, this process leads 
after a denumerable number of steps to an end. 
The discoverers of this theorem, Cantor x ) and Bendixson * 2 ) proved 
!) Mathem. Annalen 28, p. 459—471. 
2 ) Acta Matbematica 2, p. 419—427. 
