1549901369-Elements_of_Real_Analysis__Denlinger_

3.1 Neighborhoods and Open Sets 139

Definition 3.1.3 A set U ~ JR is open if Vx E U, ::le > 0 3 N,:(x) ~ U.

In words, a set U is open if and only if each of its points has a neighborhood

contained entirely in U.

u x u

~ I )

Figure 3.2

Theorem 3.1.4 Let a, b E R The intervals (a, b), (a, +oo), ( -oo, a), and

( -oo, +oo) are open sets.

Proof. (a) Consider the interval (a, b).
Case 1 (a 2: b): In this case, (a,b) = 0. Since ~x E 0, it is true that
Vx E 0 , ::le> 0 3 N 0 (x) ~ 0. Thus, 0 is open, and so (a,b) is open.
Case 2 (a < b): Let x E (a, b). Then a < x < b. Let c = min { x - a, b - x}.
Then N 0 (x) ~ (a, b). Thus, (a, b) is open.
Ni(x)
,.--"-----1

a x

Figure 3.3

b

(b) Finish the proof by considering each of the other types of intervals

given. (Exercise 1.) •

Corollary 3.1.5 Every €-neighborhood N 0 (x) is open. •

The following theorem is the basis for establishing that many other sets

are open as well. It is considered fundamental.

Theorem 3.1.6 (Open Set Theorem)

(a) 0 and JR are open.

(b) The union of any collection of open se ts is open.

(c) The intersection of any finite number of open sets is open.

Proof. (a) In proving Case 1 of Theorem 3.1.4 we proved that 0 is open.

To see that JR is open, merely observe that Vx E JR, ::le> 0 3 N 0 (x) ~ R

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