Page 67 - DMTH401_REAL ANALYSIS

Unit 3: Matric Spaces

Notes
Example: Prove that the union of two closed sets is a closed set.

Solution: Let A and B be any two closed sets. Let S = A B, we have to show that S is a closed set.
c
For this, it is enough to prove that the complement S is open
Now
c
S = (A B) = B  A = A  Bc
c
c
c
c
c
c
Since A and B are closed sets, therefore A and B are open sets. Also, we have proved in the
c
c
intersection of any two open sets is open. Therefore A  B is an open set and hence S is open.
This result can be extended to a finite number of closed sets. You can easily verify that the union
of a finite number of closed sets is a closed set. But, note that the union of an arbitrary family of
closed sets may not be closed.
For example, consider the family of closed sets given as
1 1
S = [l, 2], S = [ , 2], S = [ , 2],....
l 2 3
2 3
and in general
1
S = [ , 2]..... for n = 1, 2, 3, ....
n
n
Then,

 S = S S  S ..... S .....
1
n
=
n 1 2 3 n
= I0, 2]
which is not a closed set.
Definition: Derived Set
The set of all limit points of a given set S is called the derived set and is denoted by S'.

Example: (i) Let S be a finite set. Then $' = 

1
(ii) S = ( : n N), the derived set S' = {0}
n
(iii) The derived set of R is given by R' = R
(iv) The derived set of Q is given by Q' = R

We define another set connected with the notion of the limit point of a set. This is called the
closure of a set.
Definition: Closure of a Set

Let S be any set of real numbers (S  R). The closure of S is defined as the union of the set S and
its derived set S. It is denoted by S , Thus

S = S  S'
In other words, the closure of a set is obtained by the combination of the elements of a given set
S and its derived set S'.

LOVELY PROFESSIONAL UNIVERSITY 61