1550251515-Classical_Complex_Analysis__Gonzalez_

98 Chapter^2

Definition 2.21 The set of all limit points of a set A C S is called the
derived set of A, denoted A or A°'.
In the preceding example, A= {z: Rez;::: O}.

Definition 2.22 A set A C Sis said to be dense in itself i:ff A C A, dense

in a set B i:ff A :J B, and everywhere dense i:ff A = S.

Definition 2.23 A set A C S is said to be perfect i:ff it is both closed

and dense in itself.

Definition 2.24 A set A C S is said to be nowhere dense i:ff for each

x ES and each Ns(x) there is a neighborhood Ns1(y) C Ns(x) such that

Ns1(y) n A= 0 (8 > 0,8^1 > O).

Theorem 2.5 The following properties hold:

1. A set A C S is closed i:ff A C A.


2. A set A C S is perfect i:ff A = A.
   Definition 2.25 A metric space (S, d) is said ·to be separable if there is
   a countable subset of S that is everywhere dense~· For example, the
   real line (with the usual metric) is separable, since Q =JR, Q being the set
   of rational numbers. The space (C, d) is also separable (see Exercises 2.4,
   problem 4). ·

2.9 DISTANCE FROM A POINT TO A SET AND

DISTANCE BETWEEN TWO SETS. DIAMETER OF

A SET. BOUNDED SETS

Definition 2.26 Let (S, d) be a metric space, ACS, A-:/= 0, and x ES.

The distance from x to A, denoted d(x, A), is defined by

d( x, A) = inf { d( x, a) : a E A}

i.e., d(x, A) is the infimum (greatest lower bound) of the set of all distances

from x to the points of A.

If A and B are any two nonempty subsets of S, then the distance from

A to B (or, from B to A), denoted d(A,B), is defined by

d(A,B) = inf{d(a,b): a E A,b EB}

i.e., d(A, B) is the infimum of the set of all distances from every point of

A to every point of B.

Definition 2.27 The diameter of a set A C S, A -:/= 0 denoted .6.(A),

is defined by

6.(A) =sup {d(a, a'): a EA, a' EA}