1549901369-Elements_of_Real_Analysis__Denlinger_

144 Chapter 3 • Topology of the Real Number System

Proof. Exercise 15. •

Theorem 3.1.22 (Finite Sets)

(a) Finite sets have no interior points.

(b) Every point of a finite set A is a boundary point of A.

( c) Every point of a finite set A is an isolated point of A.

Proof. Exercise 16. •

EXERCISE SET 3.1

l. Finish proving Theorem 3.1.4.

2. Tell whether the following sets are open: (Justify your answer.)
   (a) (3, 5) U {6}
   (c) {1,2,3,4,5,6, 7,8,9}
   (e) Z
   (g) (-oo, 0) u [O, l]
   (i) { ~ : n EN}
   (k) Q


3. Prove Theorem 3.1.11 (b).


4. Prove Theorem 3.1.11 (c).

(b) (-oo,O)U(O,l)

(d) (-oo, 0) u [O, 1)

(f) (-oo, 0) U (0, l]

(h) IR - {l, 2, 3}

(j) U : n EN} U {O}

(1) Qn(O,l)

5. Prove that a set is open iff it is the union of a family of open intervals.


6. Find the interior, exterior, and boundary of each of the sets given in
   Exercise 3.1.2.


7. In Examples 3.1.10, 3.1.13, and 3.1.17 we asserted that Q^0 = 0, Qext = 0,
   and Qb = R Prove these assertions.


8. Find the interior, exterior, and boundary of the set of irrational numbers.


9. Prove that a set is open iff it contains none of its boundary points; i.e.,
   A is open¢=> AnAb = 0.


10. Give an example of a collection of bounded open intervals whose inter-
    section is [O, l]. (See Example 3.1.8.)


11. Prove Theorem 3.1.18.