1550251515-Classical_Complex_Analysis__Gonzalez_

94 Chapter^2

v y

(^0) x 0 x
(a) (b)
Fig. 2.7
Definition 2.8 For any 8 > 0 the set

N 0 (a) = {x: x E S,d(x,a) < 8}

is called an open spherical neighborhood, a 8-neighborhood, or an open ball

with center a and radius 8. In the case of a plane set of points the set

N 0 ( a) is also called a disk or a circular neighborhood.

We note that the set N 0 ( a) may contain only the center a, as in the case

of Example 1 of Section 2.5 for 8 =^1 k If S is the complex plane and d is

the usual Euclidean metric, namely, d(z, z') = lz -z'j, then the inequality

d(z,a) < 8 defines indeed a disk of center a and radius 8 (Fig. 2.7a).

However, in the metric space (C, p ), where p(z, z') = Ix - x'I + jy - y'I, the

inequality p(z,a) < 8, or Ix -ail+ jy-a21<8 (a= ai + ia2), defines the

interior of a square, as shown in Fig 2. 7b.

More generally we have:

Definition 2.9 A set N C S is called a neighborhood of a if for some

8 > 0 there is a 8-neighborhood N 0 (a) such that N 0 (a) C N (see Fig. 2.8).

y

a

(^0) x
Fig. 2.8