Principles of Mathematics in Operations Research

150 10 Basic Topology

Remark 10.2.19 Consider the following remarks on Theorem 10.2.18:

1. The equivalence of (a) and (b) is known as Heine-Ba,rel Theorem,:
   A subset E of Rk is compact if and only if it is closed and bounded.


2. (b)<&(c) holds in every metric space.


3. (c)=> (a) , (b)=> (a) hold in every metric space.
   4- (a)=>(c) , (a)=>(b) are not true in general.

Theorem 10.2.20 (Balzano-Weierstrass) Every bounded infinite subset
of Rfc has a limit point in Rk.

Proof. Let E C Kfc be infinite and bounded. Since E is bounded 3 a k-cell
/ 9 E C I. Since I is compact, E has a limit point p E I c Rk. •

Theorem 10.2.21 Let P ^ 0 be a perfect set in Rk. Then, P is countable.

10.3 The Cantor Set

Definition 10.3.1 Let
E 0 = [0,l],
E, = [0,i]U[|,l],
E 2 = [0, £] U [£, £] U [$, £] U [JM],

continue this way. Then, Cantor set C is defined as

oo

C= f]En.

n=l

Some properties are listed below:

1. C is compact.


2. C^0.


3. C contains no segment (a, (3).


4. C is perfect.


5. C is countable.

Proof (Property 3). In the first step, (|, |) has been removed; in the second
step (Jy, -p-), (T^, ^) have been removed; and so on. C contains no open in-
terval of the form (3k^^1 , 3kJ,~^2 ), since all such intervals have been removed in
the Is*,..., (n- l)s';steps.
Now, suppose C contains an interval (a,/?) where a < j3. Let a > 0 be a
constant which will be determined later. Choose n £ N 3 3~" < —^-- Let k
be the smallest integer B a < ^pr^, i.e. ^K~ < k, then k - 1 < a3' 3 ~l- Show
^ < 0, i.e. fc < ^^ k < 1 + a31=i; so show 1 + 2^LI < S^Lzl] i.e.