1549901369-Elements_of_Real_Analysis__Denlinger_

142 Chapter 3 m Topology of the Real Number System

Definition 3.1.12 (Exterior of a Set) Let A be a set of real numbers. The
exterior of A is the interior of the complement of A. In symbols,

An element of Aext is called an exterior point of A.

It is important to see the difference between the exterior of a set and the
complement of a set. The following example may help.

Examples 3.1.13 Let A = [O, 1), B = [O, 1) U {2}, C = (0, 3) U (3, 5). (See
Figure 3.6.)

0 0 1

A B

(

2 0 3

c

)

5

Figure 3.6

Then,

N = (-oo,O) U [1,oo),

B c = (-oo,O) U [1,2) U (2,oo),

cc= (-oo, OJ U {3} U [5, oo),

Additionally,

while Aext = (-oo,O) U (1,oo);

while B ext = (-oo, 0) U (1, 2) U (2, oo );

while cext = (-oo, 0) u (5, 00 ).

W = (-oo, 1) U (Q

1

(n ,n+1)), and

and !Rext = 0. D

Theorem 3.1.14 (Properties of Exterior) Let A be a set of real numbers.
Then,

(a) Aext is an open set.

(b) x is an exterior point of A iff :Jc: > 0 3 N'°(x) C Ac; i.e., x has a

neighborhood containing no points of A.

Proof. Trivial. •

Definition 3.1.15 (Boundary of a Set) Let A be a set of real numbers and
x E R We say that x is a boundary point of A if every neighborhood of x
contains at least one point of A and at least one point of Ac. The set of all
boundary points of A is called the boundary of A , and is denoted Ab.