1550251515-Classical_Complex_Analysis__Gonzalez_

2

Topology of Plane Sets of Points

2.1 INTRODUCTION

In this chapter we develop the elements of the topology of plane sets of

points as far as it is needed for complex analysis. However, the theory

is often discussed in a more general setting (that of metric spaces), since

in many cases there is no special advantage in restricting our considera-

tions to the plane. As stated in Section 0.1, we assume that the reader

has already been introduced to the basic concepts and results of point

set topology. Thus a number of definitions and some theorems that are

introduced without proofs are intended as a summary and review of the

basic material. They are also presented in order to fix the terminology

and notation to be used, as well as for easy reference in later chapters of

the book.

2.2 SOME ADDITIONAL DEFINITIONS

In Section 0.1 we introduced several concepts and notations concerning
sets: belonging to a set (a E A), not belonging (a r/. A), set inclusion
(A C B), equality of two sets (A = B), inequality (A =/:-B), union of two

sets (AU B), intersection (An B), and difference of relative complement

of B with respect to A (A..,... B). The empty set is denoted by 0 and in

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