Page 65 - DMTH401_REAL ANALYSIS

Unit 3: Matric Spaces

Notes
p – p p +
Again since p is the supremum of S, therefore, by definition p +  can not belong to S. In other
Words, p +  exceeds an infinite number of elements of A. This means that there lie an infinite
number of elements of A on the left side of the point p + .

Thus we have shown that there lies, an infinite number of elements of A on the right side of
p –  and also there ia an infinite number of elements of A on the left side of p + . What do you
conclude from this? In other words, what is the number of elements of A in between (i.e., within)
the interval ]p – , p + [. Indeed, this number is infinite i.e., there is an infinite number of
elements of A in the open interval ]p + [. Hence the interval ] p – , p +  [ contains an infinite
number of elements of A for some > 0. Since  > 0 is chosen arbitrarily, therefore every interval
]p – , p + [ has an infinite number of elements of A. Thus, every NBD-of p contains an infinite
number of elements of A. Hence p is a limit point of the set A.
This completes the proof of the theorem.

Example: (i) The intervals [0, 1], ]0,1[, ] 0, 1], [0,1[ are all infinite and bounded sets.
Therefore each of these intervals has a limit point. In fact, each of these intervals has an infinite
number of limit points because every point in each interval is a limit point of the interval.
(ii) The set [a, [ is infinite and unbounded set but has every point as a limit point. This shows
that the condition of boundedness of an infinite set is only sufficient in the theorem.
From the previous examples and exercises, it is clear that it is not necessary for an infinite set to
be bounded to possess a limit point. In other words, a set may be unbounded and still may have
a limit point. However, for a set to have a limit point, it is necessary that it is infinite.
Another obvious fact is that a limit point of a set may or may not belong to the set and a set may
have more than one limit point. We shall further study how sets can be characterized in terms of
their limit points.

3.6 Closed Sets

You have seen that a limit point of a set may or may not belong to the set. For example, consider
the set S = {xR : 0  5 x < 1}. In this set, 1 is a limit point of S but it does not belong to S. But if
you take S = {x: 0  x  1}, then all the limit points of S belong to S. Such a set is called a closed set.
We define a closed set as follows:

Definition

A set is said to be closed if it contains all its limit points.

Example: (i) Every closed and bounded interval such as [a, b] and [0, 1] is a closed set.
(ii) An open interval is not a closed set. Check Why?
(iii) The set R is a closed set because every real number is a limit point of R and it belongs to R.
(iv) The null set  is a closed set.
{ 1 }
e
(v) The set S = : n N is not a closed set. Why?
n
(vi) The set ]a, [ is not a closed set, but ] –, a] is a closed set.

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