1261 
interior or on the boundary of the cube, the value of 2) in that 
point of the X-axis which lies with MJ in one and the same space 
Of Sp, then to each system of values > — 1 and <1 for zn, ano, «… « Linn 
corresponds one and only one point of the interior or of the boundary 
of the cube, which point is a biuniform, continuous function of zn, 
Vm? se Amne Lo.w. the transformation {ep = jp}, to be represented 
by 7, , is a continuous one-one transformation of the cube with its 
boundary in itself, by which J/ passes into a countable, every where 
dense set of points M, of which no (n—1)-dimensional space parallel 
to a coordinate space contains more than one point. 
In the same way we can define a continuous one-one transfor- 
mation 7, of the cube with its boundary in itself, by which 2 passes 
into a countable, everywhere dense set of points R, of which no 
(n—1)-dimensional space parallel to a coordinate space contains more 
than one point. 
Further after the proof of theorem 7 a continuous one-one trans- 
formation 7’ of the cube with its boundary in itself exists, by which 
M, passes into Zi, so that the transformation 
Mr VAD 
possesses the properties required by theorem 8. 
We now come to a property which at first sight seems to clash 
with the conception of dimension : 
Trrorem 9. The coherence types 4" and 4 are identical. 
To prove this property, in an n-dimensional cube for which the 
rectangular coordinates of the vertices are all either 0 or 1, we con- 
sider the set J, of coherence type 1” consisting of those points 
whose coordinates when developed into a series of negative powers 
of 3, from a certain moment produce exclusively the number 1, and 
together with this we consider the set M of coherence type 4 con- 
sisting of those real numbers between 0 and 1 which when developed 
into a series of negative powers of 3”, from a certain moment pro- 
n 
duce exclusively the number —>—. The continuous Pgano represen- 
2 
tation ') of the real numbers between O and 1 on the n-dimensional 
cube with edge 1, then determines a continuous one-one represen- 
tation of M on M, establishing the exactness of theorem 9. 
That in reality theorem 9 does not clash with the conception of 
dimension, is elucidated by the remark that not every continuous 
one-one correspondence between two countable sets of points M and R, 
1) Comp. Math. Annalen 36, p. 59, and Scroenrures, Bericht über die Mengen- 
lehre I, p. 125. 
