Page 694 - Advanced_Engineering_Mathematics o'neil

674 CHAPTER 19 Complex Numbers and Functions

1. ζ is an interior point of S if there is some open disk about ζ, all of whose points are in
S. In this sense, an interior point of S is one entirely surrounded by points of S.
2. ζ is a boundary point of S if every open disk about ζ contains at least one point of S
and at least one point not in S.
A boundary point of S has points of S arbitrarily close to it and points not in S
arbitrarily close to it. In this sense, a boundary point of S is “on the edge” of S and
may or may not belong to S. A boundary point of S cannot be an interior point, and an
interior point of S cannot be a boundary point.
3. S is open (an open set) if every point of S is an interior point.
4. S is closed (a closed set)if S contains all of its boundary points.

As the following examples show, a set may be open, closed, both open and closed, or neither
open nor closed. The concepts of “open” and “closed” are not opposites—a set does not have to
be either open or closed, and not being open does not make a set closed.

EXAMPLE 19.2
Let S consist of all complex numbers z = x + iy with x ≥ 0 and y > 0, shown in Figure 19.4.
These are points (x, y) in the right quarter plane if x and y are both positive and points (0, y) on
the positive imaginary axis if x =0. Points (x,0) on the positive real axis are not in S. These facts
are indicated in the diagram by using a dashed positive real axis and a solid positive imaginary
axis.

• 1+i is an interior point of S. We can place a circle about 1+i (say of radius 1/10) containing
only points in S.
• 2i is a boundary point. Every circle about 2i contains points of S and points not in S. Because
2i is in S and is not an interior point of S, S is not open.
• 2 is also a boundary point of S, because every circle drawn about 2 contains points of S and
points not in S. However, 2 is not in S. S contains some of its boundary points but not all of
them. S is not closed. This set is neither open nor closed.

EXAMPLE 19.3
Every open disk is an open set. Every closed disk is a closed set. The open and closed disks of
radius r about z 0 have the same boundary points, namely those on the circle |z − z 0 |=r.

y

2i 1 + i S

x
2

FIGURE 19.4 The set S in Example 19.2.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 15, 2010 18:5 THM/NEIL Page-674 27410_19_ch19_p667-694