Page 63 - DMTH401_REAL ANALYSIS

Unit 3: Matric Spaces

3.5 Limit Point of a Set Notes

You have seen that the concept of an open set is linked with that of a neighbourhood of a point
on the real line. Another closely related concept with the notion of neighbourhood is that of a
limit point of a set. Before we explain the meaning of limit point of a set, let us study the
following situations:
(i) Consider a set S = [1, 2[> Obviously the number 1 belongs to S. In any NBD of the point 1,
we can always find points of S which are different from 1. For instance ]0-5, l[ is a NBD of
1. In this NBD, we can find the point 1.05 which is in S but at the same time we note that
1.05  1,

{ 1 ö
(ii) Consider another set S = n : n  N . The number 0 does not belong to this set.
÷
ø
1
Take any NBD of 0 say, ] –0.1, 0.1 [. The number = 0.05 of S is in this NBD of 0. Note that
20
0.05  0.
(i) Again consider the same set S of (ii) in which the number 1 obviously belongs to S. We can
find a NBD of 1, say ]0.9, 1.1[ in which we can not find a point of S different from 1.
In the light of the three situations, we are in a position to define the following:

Limit Point of a Set

A number p is said to be a limit point of a set S of real numbers if every neighbourhood of p
contains at least one point of the set S different from p.

Examples: (i) In the set S = [1, 2[, the number 1 is a limit point of S. This limit point belongs
{ 1 ö
to S. The set S = n : n  N has only one limit point 0. You may note that 0 does not belong to S.
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ø
(ii) Every point in Q, (the set of rational numbers), is a limit point of Q, because for every
rational number r and  > 0, i.e. ] r – 6, r +  [ has at least one rational number different from
r. This is because of the reason that there are infinite rationals between any two real
numbers. Now, you can easily see that every irrational number is also a limit point of the
set Q for the same reason.

(iii) The set N of natural numbers has no limit point because for every real number a, you can
always find  > 0 such that ]a – 6, a + [ does not contain a point of the set N other than a.
(iv) Every point of the interval ]a, b] is its limit point. The end points a and b are also the limit
points of ]a, b]. But the limit point a does not belong to it whereas the limit point b belongs
to it.
(v) Every point of the set [a, [ is a limit point of the sets. This is also true for ] –, a[.

From the foregoing examples and exercises, you can easily observe that
(i) A limit point of set may or may not belong to the set,
(ii) A set may have no limit point,
(iii) A set may have only one limit point.
(iv) A set may have more than one limit point.

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