1549901369-Elements_of_Real_Analysis__Denlinger_

3.1 Neighborhoods and Open Sets 143

Examples 3.1.16 Some boundary points:

(a) 3 and 6 are boundary points of the intervals (3, 6), (3, 6], [3, 6), and [3, 6].

(b) 0 is a boundary point of{~: n EN}. So is ~,V'n EN.

Examples 3.1.17 Let A= [O, 1), B = [O, 1) U {2}, and C = (0, 3) U (3, 5). (See

Figure 3.6.) Then

Ab= {O, 1};

Cb= {O 3 5}·

' ' '

Qb =IR;

Theorem 3.1.18 For any set A~ IR,

Bb = {O 1 2}·

' ' '

Nb =N;

IRb = 0.

(a) Ab consists of all real numbers that are in neither A^0 nor Aext.

(b) Ab = (Ac)b; that is, a set and its complement have the same boundary.

( c) A^0 , Ab, and A ext are mutually exclusive (i.e., pairwise disjoint) sets

whose union is IR.

Proof. Exercise 11. •

ISOLATED POINTS

Definition 3.1.19 A real number x is an isolated point of a set A ~ IR
if x E A and ::Jc: > 0 3 N 0 (x) contains no point of A other than x; i.e.,
N 0 (x) n A= {x}.

In words, an isolated point of A is a member of A that can be surrounded
by a neighborhood containing no other members of A.

Examples 3.1.20 Let A = [O, 1), B = [O, 1) U {2, 3, 4}, C = (0, 3) U (3, 5),
and D = U : n EN}. (See Figure 3.6.) The isolated points of these sets and
others are as follows:

Set: A B C D N Q IR

Isolated points: none {2,3,4} none D N none none

Theorem 3.1.21 Every isolated point of a set A is a boundary point of A.
The converse, however, is false.