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on one hand, and to multiply ordered sets on the other hand. In 
the first place the following theorem holds here: 
Tueorem 5. All countable sets of points lying everywhere dense 
in a cartesian R,, possess the same coherence type n”.*) 
For, to an arbitrary countable set of points, lying everywhere 
dense in A, we can construct a cartesian system of coordinates C’, 
with the property that no &,_1 parallel to a coordinate space con- 
tains more than one point of the set. If now two such sets, M and 
R, are given, then in the special case that C, and C, are identical, 
a one-one representation of M/ on RF preserving the n-fold relations 
of order as determined by C,,—= C;, can be constructed by CANTOR’s 
method cited above, only modified in as far as the “situation” 
of the points with respect to each other is determined here not by 
simple, but by n-fold relations of order. As on the grounds indicated - 
in the proof of theorem 1 this representation must also be a conti- 
nuous one, theorem 5 has been established in the special case that 
Ca and QC, are identical. From this the general case of the theorem 
ensues immediately. 
If on the other hand we have an arbitrary everywhere dense, 
countable, n-ply ordered set Z, then its n simple projections *), being 
everywhere dense, countable, and simply ordered, admit of one-one 
representations preserving the relations of order, on m countable sets 
of points lying everywhere dense on the » axes of a cartesian system 
of coordinates successively ; these 2 representations determine together 
a one-one representation preserving the relations of order, thus a 
continuous one-one representation of Z on a countable set of points, 
everywhere dense in &,. From this we conclude on account of 
theorem 5: 
TurorEM 6. All everywhere dense, countable, n-ply ordered sets 
possess the coherence type u”. 
As the n-dimensional analogon of theorem 3 the following extension 
of theorem 5 holds: 
Turorem 7. Lf in a cartesian R, be given two countable, everywhere 
dense sets of points M and R, a continuous one-one transformation 
of Ry, in itself can be constructed, by which M passes into R. 
In the special case that C,, and C, are identical, we can namely first 
construct a continuous one-one correspondence between M and R 
in the manner indicated in the proof of theorem 5, and then make 
to correspond to each point gm not belonging to M, the point gr 
having to the points of / the same (n-fold) relations of order, as ym has 
1) This theorem and its proof have been communicated to me by Prof. Boren. 
2) Comp. F. Rrrsz, lc. p. 409. 
