( 790 ) 
executed is by taking all m k s equal to 2. If then we represent the 
two elements of decomposition of the first order by p 0 and those 
of the second order by p 00 >.fA 0 *’ Pjo» t 1 ™, an ^ so on, then in this way 
the different pieces of fi are brought into a one-one correspondence 
with the different fundamental series consisting of figures 0 and 2. 
And two pieces converge to each other then, and only then, when 
the commencing segment which is common to their fundamental 
series, increases indefinitely. 
Let us consider on the other hand, in the linear continuum of 
real numbers between 0 and 1, the perfect punctual set n of those 
numbers which can be represented in the triadic system by an infi¬ 
nite number of figures 0 and 2. The geometric type of order of 
n we shall represent by £. 
Two numbers of st converge to each other then and only then, 
when the commencing segment which is common to their series of 
figures, increases indefinitely. 
So, if we realize such a one-one correspondence between the pieces 
of fi and the numbers of ar, that for each piece of ft the series of 
indices is equal to the series of figures of the corresponding number 
of ar, then to a limiting piece of a fundamental series of pieces of 
ft corresponds a limiting number of the corresponding series of 
numbers in ar, so that we can formulate: 
Theorem 3. Each perfect set of pieces possesses the geometric type 
of order £. 
For the case that the set under discussion is punctual and lies in 
a plane , this theorem ensues immediately from the following well- 
known property: 
Through each plane closed punctual set we can bring an arc of 
simple curve. 
Combining Schoenfijes’s theorem mentioned in $ 2 with theorem 
3 we can say : 
Theorem 4. Each closed set consists of two sets of pieces ', one of 
them possesses, if it does not vanish, the geometric type of order 5, 
and the other is denumerable. 
$ 4. 
The groups which transform the geometric type of order £ in itself. 
Just as spaces admit of groups of continuous one-one transforma¬ 
tions, whose geometric types of order *) are again spaces, namely 
i) In this special case formerly called by me •Parametermannigfaltigkeiten” 
Comp. Mathem. Annalen 67, p. 247. 
