Page 66 - DMTH401_REAL ANALYSIS

Real Analysis

Notes You may be thinking that the word open and closed should be having some link. If you are
guessing some relation between the two terms, then you are hundred per cent correct. Indeed,
there is a fundamental connection between open and closed sets. What exactly is the relation
between the two? Can you try to find out? Consider, the following subsets of R:

(i) ]0, 4[
(ii) [–2, 5]

(iii) ]0, = [
(iv) ] – > 6].

The sets (i) and (iii) are open while (ii) and (iv) are closed. If you consider their complements,
then the complements of the open sets are closed while those of the closed sets are open. In fact,
we have the following concrete situation in the form of following theorem.
Theorem 2: A set is closed if and only if its complement is open.

c
Proof: We assume that S is a closed set. Then we prove that its complement S is open.
c
c
C
To show that S is open, we have to prove that S is a NBD of each of its points. Let xS . Then, x
c
S  x  S. This means x is not a limit point of S because S is given to be a closed set. Therefore,
there exists a  > 0 such that ]x – , x + [ contains no points of S. This means that ]x – , x + [ is
c
c
c
contained in S . This further implies that S is a NBD of x. In other words, S is an open set, which
proves the assertion.
c
Conversely, let a set S be such that its complement S is open. Then we prove that S is closed.
To show that S is closed, we have to prove that every limit point x of S belongs to S. Suppose
C
x S, Then x S .
c
C
This implies that S is a NBD of x because S is open. This means that there exists an open interval
]x – , x + [, for some 6 > 0, such that
]x – , x + [S c
In other words, ]x – , x + [ contains no point of S. Thus x is not a limit point of S, which is a
contradiction. Thus our supposition is wrong and hence, x S is not possible. In other words, the
(limit) point x belongs to S and thus S is a closed set.
Note that the notions of open and closed sets are not mutually exclusive. In other words, if a set
is open, then it is not necessary that it can not be closed. Similarly, if a set is closed, then it does
not exclude the possibility of its being open. In fact, there are sets which are both open and
closed and there are sets which are neither open nor closed as you must have noticed in the
various examples we have given in our discussion. For example the set R of all the real numbers
is both an open sets as well as a closed set. Can you give another example? What about the null
set. Again Q, the set of rational numbers is neither open nor closed.

Task Give examples of two sets which are neither closed nor open.

We have discussed the behaviour of the union and intersection of open sets. Since closed sets are
closely connected with open sets, therefore, it is quite natural that we should say something
about the union and intersection of closed sets. In fact, we have the following results:

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