1262 
lying everywhere dense in Ru, admits of an extension to a continuous 
one-one transformation of R,-in itself. If e.g. the set of the rational 
points of the open straight line is submitted to the continuous one- 
one transformation 2’ = , this transformation does not admit of 
Tt —& 
an extension to a continuous one-one transformation of the open 
straight line in itself. 
A more characteristic example, presenting the property moreover 
that in no partial region an extension is possible, we get as follows: 
Let ¢, denote the set of those real numbers between OQ and 1 of 
which the development in the nonal system from a certain moment 
produces exclusively the digit 4, ¢, the set of the finite ternal fractions 
between O and 1. Let 7’ denote a continuous one-one transformation 
of the set of the real numbers between O and 1 in itself, by which 
t, passes into ¢, + ¢,, thus a part f, of ¢, into ¢,, and a part ¢, of t, 
into ¢,. By a Peano representation 7, the sets ¢,, ¢,, ¢,, t, successively 
pass into countable sets of points s,, s,, 53, 8,, lying everywhere dense 
within a square with side unity, and, so far as are concerned, s,, s,, and 
s, containing no points of the boundary of this square. The continuous 
one-one representation T of t, on ¢, now determines a continuous one- 
one representation T,=T,TT,—' of s, on s,, not capable of an 
extension to a continuous one-one representation of the interior of the 
square in itself. For, if such an extension would exist, it would be, 
for each set of points in the interior of the square, the only possible 
continuous extension of 7. For s,, however, 7’,7’7,—! furnishes 
itself such a continuous extension, which we know to be not a one- 
one representation. 
The conception of dimension can now be saved, at least for the 
everywhere dense, countable sets of points, by replacing the notion 
of coherence type by the notion of geometric type*). Two sets of 
points will namely be said to possess the same geometric type, if a 
uniformly continwous one-one correpondence exists between them. 
And it is for uniformly continuous representations that the following 
property holds: 
Tueorem 10. Every uniformly continuous one-one correspondence 
between two countable sets of points M and R, lying everywhere dense 
in an n-dimensional cube, admits of an extension to a continuous 
one-one transformation of the cube with its boundary in itself. 
1) For closed sets the two notions are equivalent. For these they were intro- 
duced formerly under the name of geometric type of order, these Proceedings XII, 
p. 786. 
