The Real And Complex Number Systems

(b) If x is an accumulation point of S, then x is an accumulation point of at least one set
AinF.
Proof: No! For example, Let S R n ,andF be the collection of sets consisting of a
single point x R n . Then it is trivially seen that S AF A.Andifx is an accumulation
point of S, then x is not an accumulation point of each set A in F.
3.16 Prove that the set S of rational numbers in the inerval 0, 1 cannot be
expressed as the intersection of a countable collection of open sets. Hint: Write
S x 1 , x 2 ,..., assume that S k k1 S k , where each S k is open, and construct a
sequence Q n of closed intervals such that Q n1 Q n S n and such that x n Q n .
Then use the Cantor intersection theorem to obtain a contradiction.
Proof: We prove the statement by method of contradiction. Write S x 1 , x 2 ,..., and
assume that S k k1 S k , where each S k is open.
Since x 1 S 1 , there exists a bounded and open interval I 1 S 1 such that x 1 I 1 .
Choose a closed interval Q 1 I 1 such that x 1 Q 1 .SinceQ 1 is an interval, it contains
infinite rationals, call one of these, x 2 .Sincex 2 S 2 , there exists an open interval I 2 S 2
and I 2 Q 1 . Choose a closed interval Q 2 I 2 such that x 2 Q 2 . Suppose Q n has been
constructed so that
1. Q n is a closed interval.
2. Q n Q n1 S n1 .
3. x n Q n .
Since Q n is an interval, it contains infinite rationals, call one of these, x n1 .Since
x n1 S n1 , there exists an open interval I n1 S n1 and I n1 Q n . Choose a closed
interval Q n1 I n1 such that x n1 Q n1 .So,Q n1 satisfies our induction hypothesis, and
the construction can process.
Note that
1. For all n, Q n is not empty.
2. For all n, Q n is bounded since I 1 is bounded.
3. Q n1 Q n .
4. x n Q n .
Then n n1 Q n by Cantor Intersection Theorem.
Since Q n S n , n n1 Q n n n1 S n S. So, we have
S n n1 Q n n n1 Q n
which is absurb since S n n1 Q n by the fact x n Q n . Hence, we have proved that
our assumption does not hold. That is, S the set of rational numbers in the inerval 0, 1
cannot be expressed as the intersection of a countable collection of open sets.
Remark: 1. Often, the property is described by saying Q is not an G set.
2. It should be noted that Q c is an G set.
3. For the famous Theorem called Cantor Intersection Theorem, the reader should
see another classical text book, Principles of Mathematical Analysis written by Walter
Rudin, Theorem 3.10 in page 53.
4. For the method of proof, the reader should see another classical text book, Principles
of Mathematical Analysis written by Walter Rudin, Theorem 2.43, in page 41.