152 Chapter 3 • Topology of the Real Number System
- Tell whether the following sets are open, closed, both, or neither:
(a) (3, 5) U {6}
(c) {1,2,3,4,5,6, 7,8,9}
(e) Z
(g) (-oo, 0) U [O, 1]
(i) U: n EN}
(k) Q
(b) (-oo, O) u (0, 1)
(d) (-oo, 0) U [O, 1)
(f) (-oo, 0) U (0, 1]
(h) IR - {1, 2, 3}
(j) { ~ : n E N} U {O}
(1) Q n (O, 1)- Prove Theorem 3.2.4 (a) and (c). [Hint: Use the open set theorem and,
for (c), use de Morgan's law.] - Find all the cluster points of each set given in Exercise 3.2.3.
- Give an example of a collection of bounded closed intervals whose union
is unbounded and not closed. - Suppose A -1- 0 and A is bounded above. Is sup A necessarily a cluster
point of A? Prove that if sup A tf. A , then it is a cluster point of A. State
and prove analogous results for inf A. - Prove that every nonempty closed set that is bounded above contains a
maximum element, and every nonempty closed set that is bounded below
contains a minimum element. - Prove Lemma 3.2.9.
10. Prove Corollary 3.2.10.
Prove Coroll ary 3.2.12.
Prove that
(a) If A is open and Bis closed, then A - Bis open;
(b) If A is closed and B is open, then A - B is closed.
Finish proving Theorem 3.2.15.
Prove that A= AU Ab. [Show how this follows from Theorem 3.2.17.]
Suppose A is a nonempty set of real numbers. Prove that
(a) If A is bounded above, then sup A EA.
(b) If A is bounded below, then inf A EA.