1958 
is evident. by choosing for 74, the point of A with the smallest 
index, having with respect to 7,,7,,...7) the same situation, as m,41 
has with respect tom, mi, mi. The correspondence constructed in 
this way, is at the same time continuous ; for, the limiting point 
relations depend exclusively on the relations of order, as a point m 
is then and only then a limiting point of a sequence f, if each 
interval containing m contains an infinite number of points of f. 
The above proof shows at the same time the independence of the 
coherence type 4 of the linear continuum. For, after Canror it leads 
also to the followirg more general result: 
Tunorem 2. All everywhere dense, countable, simply ordered sets 
possess the coherence type n°) 
Theorem 1 may be extended as follows: 
Turorem 3. Jf on the open straight line be given two countable, 
everywhere dense sets of points M and R, a continuous one-one 
transformation of the open straight line in itself can be constructed, 
by which M passes into R. 
In order to define such a transformation, we first by CANTOR’s 
method construct a continuous one-one representation of M on R.- 
Then the order of succession of the points of M is the same as the 
order of succession of the corresponding points of A. We further 
make to correspond to each point gm of the straight line not be- 
longing to M, the point gr having to the points of R tbe same 
relations of order, as gm has to the corresponding points of M. In 
this way we get ‘a one-one transformation of the straight line in 
itself, preserving the relations of order. On the grounds indicated in 
the proof of theorem 1 this transformation must also be a continu- 
ous one. 
Analogously to theorem 3 is proved: 
Tunorem 4. If within a finite line segment be given two countable, 
everywhere dense sets of points M and Rk, a continuous one-one trans- 
formation of the line seyment, the endpoints included, in itself 
can be constructed, by which M passes into R. 
We shall now treat the question, to what extent the theorems 
1, 2, 3, and 4 may be generalized to polydimensional sets of points 
1) The possibility of a definition founded exclusively on relations of order, shewn 
by Canror not only for the coherence type +, bul likewise for the coherence type 
3 of the complete linear continuum, holds also for the coherence type & of the 
perfect, punctual sets of points in Rn» (comp. these Proceedings XIl, p. 790). As 
is easily proved, this coherence type belongs to all perfect, nowhere dense, simply 
ordered sets of which the set of intervals is countable (an “interval” is formed 
here by each pair of elements between which no further elements lie). 
