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Mathematics. — “Continuous one-one transformations of surfaces 
in themselves”. (Brd Communication ')). By Dr. L. B. J. Brouwer. 
(Communicated by Prof. D. J. Korruwre). 
(Communicated in the meeting of December 24, 1910). 
In the second communication several points of the argumentation 
were indicated, but in short. We shall now treat some auxiliary 
theorems, of which the proof is necessary for a complete develop- 
ment of the theory. 
s 1. 
Definitions and lemmas. 
On a surface we understand by a jinite continuum a coherent set 
of points containing all its limiting points, and in which each funda- 
mental series of points possesses a limiting point. 
By a continuum we understand a coherent set of points containing 
all its limiting points and containing for every two of its points a 
finite continuum joining those two points. 
A finite continuum determining only one rest region we shall 
call a circular continuum, if that rest region possesses for analysis 
situs the character of a rest region of a trema. 
A continuum determining only one rest region, which is for 
analysis situs equivalent to the surface itself, we call a paraboli 
continuum °). 
A circular or a parabolic continuum together with a certain vicinity 
of it allows of a continuous one-one representation on a region 
of a Cartesian plane. There the circular continuum then lies entirel 
in a finite region, the parabolic continuum extends to the infinite 
Both determine in the Cartesian plane only one rest region and 
possess there a single circumference of accessible points, which lie 
in eyelie order for the circular continuum and in linear order for 
the parabolic continuum. 
1) See these Proceedings Vol. XI, page 788, Vol. XII, p. 286. Compare also 
the abstract: “Ueber eineindeutige, stetige Transformationen von Flächen in 
sich” (Mathem. Annalen, Vol. 69, page 176), where the endresult of these researches 
is formulated. 
*) I do not maintain the term “open system of curves”, which I used in the 
preceding communication (p. 294 sq.) for a nowhere dense circular or parabolic 
continuum, 
50 
Proceedings Royal Acad. Amsterdam. Vol, XII, 
