1549901369-Elements_of_Real_Analysis__Denlinger_

168 Chapter 3 • Topology of the Real Number System

of some number in [O, 1] since it consists of all O's and l's. Moreover, there is

only one such number. Let f(x) =this number. Thus, \:/x EC,

f(x) = dec~mal representa.tion_ results when all

{

the real number in [O, 1] whose base-2 }

2's m the nontermmatmg ternary expan-

sion of x are replaced by 1 's.

It is easy to see that f : C -+ [O, 1] is a 1-1 correspondence. •

Remark: While Example 3.4.11 shows that the Cantor set contains points

other than endpoints of "removed intervals,'' Theorem 3.4.12 allows us to show

the far more remarkable fact that the Cantor set contains uncountably many

points that are not "endpoints," whereas there are only countably many such

endpoints. (See Exercise 6.)

Definition 3.4.13 A set A of real numbers is perfect if A' = A; that is, A

consists of all its cluster points.

Examples 3.4.14 The following sets are perfect:

(a) 0 and IR;

(b) ( -oo, a ] and [a, +oo), for any real number a;

( c) [a, b], for any real numbers a, b with a < b.

Proof. Exercise 8. D

Theorem 3.4.15 The Cantor set is a perfect set.

Proof. Let C denote the Cantor set.

(a) C' <:;;; C , since C is closed.

(b) To prove that C <:;;; C', choose any x E C. Let c >. Then 3n E N 3

1

3

n < c, and by definition of C , x E Cn. Then x belongs to exactly one of the

1

2n+l disjoint closed intervals of length

3

n comprising Cn; call it In = [an, bnl·

!"

a,, x b,,

Figure 3.17