By Stanislaw Saks

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Are examples of simply connected regions. Annuli, and in particular - an­ nular neighbourhoods, are doubly connected regions. The open plane, after removing from it the sequence of points 1,2, • . ,n, . . , becomes an infini tely connected region. 14) Every component of an open set G not separating the pT,a,ne is · a simply connected region. Pr o o f. 5 (4°) the set of these components 32 INTRODUCTION. Theory of sets. is at most denumerable. Let I Hn} and let H be any one of these components denote the sequence of the remaining components.

In the sequel, when we speak of open circles, we shall always tacitly assume that they are circles of positive radii and hence non­ empty sets. On the other hand, where circles of radius 0 can appear, this will be explicitly stated. If a is a point of the plane and r11r2 real numbers (finite or infinite} such that O�r1

E-sided derivatives). If two curves C and I' are not essentially different, then we write C=I'. It is evident that for every curve C we have C=C; besides, that if C1 =C1, then also C1=C1; finally, if C1 =C2 ·and 02=C1, then 01=0 1• In the sequel, curves which are not essentially different will be frequently denoted by the same letters. We shall distinguish only those properties of curves which are common to a curve C0 and to every curve C= 00• Concerning these properties we shall they do not depend on the parametric representation of the curve.

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