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16 £. f lies everywhere dense on k. Then g must also lie everywhere 
dense on k. 
For in the opposite case the arcs of curve k } k', k ", etc. form an 
arc of curve B, in whose circumference the domains d, d', d", etc. 
leave free a part everywhere dense in B, whilst each arc $*) coheres 
with &"" 1 ) and but with none of the other. This however 
is impossible, as we have shown above, that the last and the first 
domain must cohere with each other. 
But if g lies everywhere dense on k, k is a part of k', and again 
we are in case II. 
II. Z' is a part of Z. We then, and also in the preceding cases 
reduced to this, have an open system of curves') S, having as its 
image S' a part of itself. The image S" of S' is again a part of 
S', etc. If S (“) is the set common to all sets S^, it is an open 
system of curves i, invariant for the transformation. 
From the commencement of the domain construction according to 
fig. 2 is evident, that the two sets 7 \ in which u and u' meet each 
other, and therefore also the two invariant sets i, lie isolated from 
each other. 
At the commencement of the domain construction we have pre¬ 
supposed, that the two arcs of curve of fig. 1 have no endpoint in 
common. If that is the case, the method needs but slight modifications, 
giving no difficulties. We then start not with common arcs of simple 
curve accessible for both O arid O', but only with common points 
accessible for both 0 and O', which is sufficient too. 
b This means a perfect, coherent, nowhere dense set, determining in a region 
of the connection of the inner region of a circle or parabola only owe rest region. 
