1260 
to the corresponding points of M/. In this way we get a one-one transfor- 
mation of R, in itself preserving the relations of order as determined 
by Cy, = C;. As on the grounds indicated in the proof of theorem 1 
this transformation is also a continuous one, theorem 7 has been 
established in the special case that C,, and C, are identical. From 
this the general case of the theorem ensues immediately. 
The n-dimensional extension of theorem 4 runs as follows: 
Tunorrm 8. Jf within an n-dimensional cube be given two coun- 
table, everywhere dense sets of points M and R, a continuous one-one 
transformation of the cube, the boundary included, in itself 
can be constructed, by which M passes into R. 
The proof of this theorem is somewhat more complicated than 
those of the preceding ones. We choose in Zi, such a. rectangular 
system of coordinates that the coordinates z,, 7, .... 2, of the 
cube vertices are all either +1 or — 1, and for p=1,2,...n 
successively we try to form a continuous transition between the 
(n—1)-dimensional spaces «,=— 1 and a—=-+1 by means of a 
onedimensional continuum sj, of plane (n—1)-dimensional spaces 
meeting each other neither in the interior nor on the boundary of 
the cube, and containing each at most one point of M. In this 
” 
we succeed as follows: Let S= © ap vp = c be a plane (n—1)-dimen- 
pst 
sional space containing 9 straight line parallel to a line fF, joining 
two points of) 7) and throteh each point (pp MS == 7, = 
ip = 6, f= tia ss) = a 0) Tet nedap van (a —1)-dimener 
onal space: zp + e(1 — a?) S=a- eapa (1 — a’); in this way we 
get a continuous series 6, of plane (n—1)-dimensional spaces, and 
we can choose a magnitude e, with the property that for e <e, two 
arbitrary spaces of 6, meet each other neither in the interior nor on 
the boundary of the enbe. As further an (n—1)-dimensional space 
belongs to at most one 6, thus a line fF, is contained in an (n—1)- 
dimensional space belonging to 6, for at most one value of e, and the 
lines f,, exist in countable number only, it is possible to choose a 
suitable value for e< e, with the property that no space of 6, con- 
tains a line Ff, i.o.w. that 6, satisfies the conditions imposed to s,,-. 
If for each value of p we choose out of s,, an arbitrary space, 
then these m spaces possess one single point, lying in the interior of 
the cube, in common. For, by projecting an arbitrary space of sj 
together with the sections determined in it by Sn2, Sn3,.--- Simm into 
the space a, = 0, we reduce this property of the „-dimensional cube 
to the analogous property of the (n—1)-dimensional cube. So if we 
introduce as the coordinate ap of an arbitrary point A lying in the 
