1549901369-Elements_of_Real_Analysis__Denlinger_

3.1 Neighborhoods and Open Sets 141

INTERIOR, EXTERIOR, AND BOUNDARY

Definition 3.1.9 (Interior of a Set) Let A be a set of real numbers. A real

number x is said to be an interior point of A if 3 c: > 0 :::i Ne; ( x ) ~ A. That is,

an interior point of A can be surrounded by a neighborhood contained entirely

in A.

The interior of A is the set

A^0 = {x : x is an interior point of A}.

Examples 3.1.10 Let A = [O, 1), B = [O, 1) U {2}, and C = (0, 3) U (3, 5).

Then

[ )

0 1

A

(a) A^0 =(0,1)

(c) C^0 = C

(e) JRO =JR

0 1

B

2

(b) B^0 =(0,1)

(d) N° = 0

(f) Q^0 = 0.

(

0

Figure 3.5

3

c

)

5

Theorem 3.1.11 (Properties of Interior) Let A be a set of real numbers.
Then ,

(a) A^0 = U{all open subsets of A};

(b) A^0 is th e largest open subset of A , in the sense that A^0 is open and _if U

is an open subset oiA,_ then U ~ A^0 ;

• ~~
  ( c) A is open <=> A = A^0.

Proof. (a) Part 1: Let x E A^0. By definition, :Jc:> 0 :::i Nc;(x) ~A. Thus
x is a member of an open subset of A ; namely, Nc;(x ). Hence, x EU {all open
subsets of A}. Therefore, A^0 ~ U {all open subsets of A}.
Part 2: Let x EU {all open subsets of A}. Then, 3 an open subset U of A
such that x EU. Since U is open , :Jc:> 0 :::i Nc;(x ) ~ U ~A. Thus, x E A^0 •
Therefore, U {all open subsets of A} ~ A^0 •
(b) Exercise 3.
(c) Exercise 4. •