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m, for w constant the only limiting element of the sequence fm}, 
then each of the latter sequences contains such an end segment {1,5}, 
that an arbitrary sequence of elements m,), for which u continually 
increases, possesses m as its only limiting element. 
The sets of points of an n-dimensional space form a special case 
of normally connected sets. 
Another special case we get in the following way: In an n-ply 
ordered set’) we understand by an interval the partial set formed. 
by the elements w satisfying for @e <n different values of 7a relation 
of the form 
ng i i 
beun of bi <u OR a ae 
we further define an element m to be a limiting element of a sequence 
f, if each interval containing m, contains elements of / not identical 
to m, and the given set to be everywhere dense, if none of its inter- 
vals reduces to zero. Then the everywhere dense, countable, n-ply 
ordered sets which will be considered more closely in this paper, 
likewise belong to the class of normally connected sets. 
A representation of a normally connected set preserving the limiting 
element relations, will be called a continuous representation. 
If of a normally connected set there exists a continuous one-one 
representation on an other normally connected set, the two sets will 
be said to possess the same coherence type. 
One of the simplest coherence types is the type 4 already intro- 
duced by Cantor’). From a proof of Cantor follows namely : 
Taeorem 1. All countable sets of points lying everywhere dense on 
the open straight line, possess the same coherence type u. 
The proof is founded on the following construction of a one-one corre- 
spondence preserving the relations of order, between two sets of points 
MS mn and. Birr. dof the. class considered +) Ta 
r, CANTOR makes to correspond the point m,; to r, the point m;, 
with the smallest index, having with respect to m, the same situation 
(determined by a relation of order), as r, has with respect to 7,; to 
r, the point m;, with the smallest index, having with respect to m, 
and m;, the same situation (determined by two relations of order), 
as r, has with respect to 7, and r,; and so on. That in this way 
not only all points of #, but also all points of M have their turn, 
1.o.w. that if among m,,mj,,-..mi, appear m,,m,,...m,, but not 
m,41, there exists a number 5 with the property that m,41 = Mi» 
1) Comp. F. Riesz, Mathem. Annalen 61, p. 406. 
2) Mathem. Annalen 46, p. 504. 
