1549901369-Elements_of_Real_Analysis__Denlinger_

150 Chapter 3 111 Topology of the Real Number System

Examples 3.2.16 Let A= [O, 1), B = [O, 1) U {2}, and C = (0, 3) U (3, 5). (See

Examples 3.1.10 and 3.1.13.) Then,

A= [O, l] ; B = [O, l] u {2}; C = [O, 5]; N = N; Q =JR; iR: = R

[ )( )

0 0 2 0 3 5

Figure 3.8

Theorem 3.2.17 Let A'= the set? of all cluster points of A. Then A= AUA'.

(Thus, every point of A is either a point of A or a cluster point of A.)

Proof. (a) First we prove that As;;; AU A'. [i.e., x EA=> x E AU A'.]

We prove the contrapositive. Suppose x tf: A U A'. Then x tf: A and x tf: A', so

:le > 0 3 N 0 (x) contains no points of A. Then N 0 (x) s;;; Ac, so A s;;; N 0 (x)c.

Now N 0 (x)c is a closed set containing A, since N 0 (x) is open. But x t/: N 0 (x)c.

Thus x tf: n{all closed subsets containing A}; i.e., x tf: A. Therefore, As;;; AUA'.

(b) Next we prove that AU A's;;; A. Let x EAU A'.

Case 1 (x EA): Then x EA since As;;; A.

Case 2 (x E A'): Then x is a cluster point of A, so by definition

of cluster point, x is a cluster point of A. But A is closed, so by

Theorem 3.2.8, x E A.

In either case, x EA. Therefore, AU A's;;; A.

(c) By (a) and (b) together, A= AU A'. •

SEQUENTIAL CRITERIA

Theorem 3.2.18 (Sequential Criterion for Cluster Points) x is a clus-

ter point of a set A iff :J sequence {an} of points of A other than x, such that

an_, x.

Proof. (a) ( =>): Suppose x is a cluster point of A. Then , by definition,
Vn E N, the neighborhood N-;; (x) contains a point an E A other than x.

Consider the sequence {an}· Note that Vn EN, an E Ni(x), so Ian - xi<~-

By the squeeze principle, an _, x. n

2. The set A' is often called the derived set of A.