for any limit region, to which the region G may be extended, and 
likewise for its image region. 
We now extend G in such a way, i. e. we replace it in such a 
way by a region in which it is contained, that, when G' has obtained 
the corresponding extension, the extended regions G and G' still lie 
outside each* other. We repeat this extending process so often, until 
we get limit region, and this will be the case after a denumerable 
number of extensions. 
If possible, we then, after an indefinitely small modification of the 
boundary, execute an enlargement of this limit region; by this its 
property of being a limit region will in general be lost, but can be 
regained by a denumerable number of new extensions. This new 
limit region we again try to enlarge, and in this way go on, until 
by a denumerable number of operations a transformation domain 
0 is obtained. 
If in the surface no region exists, which at once with all points 
of its boundary is invariant for the transformation, the domain 0 
can at most determine two rest regions, namely a rest region R v 
in which O' lies, and a rest region R„ identical to R l or not, in 
whose image region lies O. 
If namely a third rest region G 8 existed, G t as well as G\ would 
be free of 0, as well as of O'. Let P be an arbitrary point on 
the boundary of G s not coinciding with its image P'. Let us con¬ 
struct about P and P' simple closed curves, chosen as small as one 
likes, which are each other’s image and bound regions sx and n', 
then ni ( 0 , G Si st) and ni (O', G' t ,n') each contain one of a pair ot 
regions, which are each other’s image, and which one can make to 
contain of 0 and G 3 resp. of O' and G\ as closely as one likes 
approximating x ) partial regions, to which 0 and O' might be enlarged, 
but this would clash With the property of a transformation domain. 
The same reasoning holds with a slight modification for the supposition 
that G z and G\ coincide. 
We shall say, that a region of a surface does not break the con¬ 
nection of the surface, if it determines only one rest region, possessing 
for analysis situs the character of a rest region of a trerna. Then 
of course the region itself is singly connected. 
We shall now assume, that O does not break the connection of 
the surface. Then it is bounded by a closed curve *) K, which 
on account of the commencement of the construction according ,0 
fig. 2 is a non-singular closed curve , and therefore does allow of 
b Schonflies, Bericht fiber die Mengenlehre II, p. 104 sqq. 
s ) Id., ibid., p. 118 sqq. 
