1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

1.6 • THE TOPOLOGY OF COMPLEX NUMBERS 43

y

-2-~x

FigW'e 1.26 The annulus A= {z : I < lzl < 2} is a connected set.

in A, as shown in Figure 1.26. The set B = {z: lz + 21 <1 or lz - 21 < l }

consists of two disjointed disks. We leave it as an exercise for you to show that
the set is not connected, as shown in Figure 1.27.
We call a connected open set a domain. In the exercises we ask you to show

that the open unit disk D 1 (0) = {z: lzl < l} is a domain and that the closed

unit disk D 1 (O) = {z: lzl ~ l} is not a domain. The term domain is a noun and
is a type of set. In Chapter 2 we note that it also refers to the set of points on
which a function is defined. In the latter context, it does not necessarily mean
a connected open set.

y

• l

FigW'e 1.2 7 The set B = { z : lz + 21 < I or lz - 21 < I} is not a. connected set.