By Stanislaw Saks

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In the plane E, however, all sets are compact. 3) If P and Q are sets c"losed in the, plane, then there exist two points peP and qeQ such that § 8 23 The plane. 4) e(p ,q) =e(P , Q). The distance between two closed sets is therefore equal to zero if and only if these sets have a common point. Pro o f. The theorem is obvious when fore assume that the sets at least, e. g. the set {p } .. } P, P and Q P·Q¥:-0. We can there­ are disjoint. Hence one of them cx:i. )-e(P,Q). 2) we can extract from this sequence convergent subsequence {p,.

To the definition given above the distance between two points in the plane is infinite if and only if one of these points is 0 and the other oo . By the d i stance e(a,P) of the point a from the set P in the e(a,p), when pt:P; and by the distance e(P ,Q) between two sets P and Q the lower bound of the numbers e(p,q), when pt:P and qt:Q. Finally, by the plane E we mean the lower bound of the numbers 2* INTRODUCTION. Theory of sets. 20 diameter of a set we mean the upper bound of the distances between any two points of this set.

Z are call the • • • neighbourhood hood with centre (z ,z2). 1 and z(l)' points of the plane. ,z�m>) in the space Em we shall mean the lar­ k gest of the numbers e(z<�>,z� >) for k=l,2, ... ,m. 1 (a) it follows immediately that the relation limzk = z in the k space Em is equivalent to the relation e(zk ,z)�O. By the distance CHAPTER I FUNCTIONS OF A COMPLEX VARIABLE § 1. Continuous functions. In this section we shall establish fundamental definitions and notations concerning functions of one and of several variables.

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