Mathematics. — “Note on linear homogeneous sets of points’. By 
Dr. B. P. Haatmerer. (Communicated by Prof. L. B. J. Brouwer). 
(Communicated at the meeting of October 25, 1919). 
We shall call a linear set of points « homogeneous in the interval 
AB, if its subset, interior to an arbitrary sub-interval, allows of a 
uniformly continuous one-one representation on the subset of ar 
interior to 4B *). 
If the set z is everywhere dense in the interval 4B*®), each of 
these representations determines a continuous one-one correspondence 
between the entire linesegments. As will be shown, we may in 
this case, assume the correspondences, postulated for a homogeneous 
set of points, to leave relations of order unaltered. 
Let CD be a sub-interval of AB (possibly identical to AB) and 
E a point between C and D. We consider the following possibilities : 
1. For every system of points C, D, and ZE the representation 
of the interval CD on CE leaves relations of order unaltered. 
2. This is not the case. 
First case. Suppose a representation of 4B on FH has to be 
effected (order from left to right A, F,H, B). According to the 
assumption both AB and FH can be represented on AH with 
unaltered relations of order, hence AL on FH in the same way. 
Second case. The assumption postulates the existence of an interval 
CD which can be represented on its sub-interval CE with inverted 
relations of order. Considering this representation is a continuous 
one-one correspondence between entire linesegments, it follows from 
the Depekinp axiom that a point P exists (not necessarily belonging 
to the set 2), which corresponds to itself. This however establishes 
the possibility of representing the part of a interior to CD on 
itself with inversion of order-relations. It follows that the part of zr 
interior to an arbitrary sub-interval of AB, possesses this same 
') An analogous definition has been given by Hausporrr for ordered sets, 
Grundz. der Mengenlehre p. 173. Wor linear sets of points Brouwer has introduced 
the following more extensive definition: a linear set of points + is homogeneous 
in the interval AB if for each couple PQ of its points interior to AB, there exists 
a continuous one-one transformation of the interval AB in itself, such that zr 
passes into itself and the point P into the point Q. These Proceedings XX, p. 1194. 
*) Which obviously is the case if + has any points inside AB. 
45 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
