1549901369-Elements_of_Real_Analysis__Denlinger_

146 Chapter 3 • Topology of the Real Number System

3.2 Closed Sets and Cluster Points

The notion of "closed set" is closely related to the notion of "open set,'' but

the relation is not exactly what one might expect. Be forewarned that "closed"

does not mean "not open." Unlike doors and restaurants, sets that are open

might also be closed, and sets that are not open are not necessarily closed! Pay

careful attention to the following definition.

Definition 3.2.1 A set of real numbers is closed if its complement is open.

That is, A is closed {::} Ac is open.

Corollary 3.2.2 \:/a,b E JR, the following sets are closed: 0, {a}, (-oo,a],

[a,b], [a, + oo), JR.

Proof. Exercise 1. •

Note 1: if a< b, then the sets (a, b] and [a, b) are neither open nor closed.

Proof. Exercise 2. •

Theorem 3.2.3 The boundary of any set is closed.

Proof. Let A~ JR. By Theorem 3.1.18, (Ab)c = A^0 UAext, and by Theorems

3.1.11 and 3.1. 14 , both A^0 and Aext are open sets. Thus, A^0 U Aext is open;

i.e., (Ab)c is open. Therefore, by definition, Ab is closed. •

Theorem 3.2.4 (Closed Set Theorem)

(a) 0 and JR are closed.

(b) The intersection of any collection of closed sets is closed.

( c) The union of any finite number of closed sets is closed.

Proof. (a) Exercise 4.

(b) Let e be any collection of closed sets. Recall de Morgan's law for col-

lections of sets (see Appendix B , Theorem B.1.10):

(n er= ( n A)c = LJ Ac.

AEC AEC

Since each set A Ee is closed, each A c is open. Thus, LJ Ac is the union
AEC
of a collection of open sets, and so by the open set theorem, is open. That is,
(n er is open. By Definition 3.2.1, this means that n e is closed.

(c) Exercise 4. •