Page 68 - DMTH503_TOPOLOGY

Topology Sachin Kaushal, Lovely Professional University

Notes Unit 6: Closed Sets and Limit Point

CONTENTS
Objectives
Introduction

6.1 Closed Sets
6.2 Limit Point
6.2.1 Derived set

6.3 Summary
6.4 Keywords
6.5 Review Questions
6.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Define closed sets;
 Solve the problems related to closed sets;

 Understand the limit points and derived set;
 Solve the problems on limit points.
Introduction

On the real number line we have a notion of ‘closeness’. For example each point in the sequence
1..01..001..0001..00001.. is closer to 0 than the previous one. Indeed, in some sense 0 is a limit
point of this sequence. So the interval (0, 1] is not closed as it does not contain the limit point 0.
In a general topological space me do not have a ‘distance function’, so we must proceed differently.
We shall define the notion of limit point without resorting to distance. Even with our new
definition of limit point, the point 0 will still be a limit point of (0, 1]. The introduction of the
notion of limit point will lead us to a much better understanding of the notion of closed set.

6.1 Closed Sets

A subset A of a topological space X is said to be closed if the set X-A is open.

Example 1: The subset [a, b] of R is closed because its complement
R  [a, b] = (, a)  (b, + ) is open.

Similarly, [a, + ) is closed, because its complement (, a) is open. These facts justify our use of
the terms “closed interval” and “closed ray”. The subset [a, b) of R is neither open nor closed.

Example 2: In the discrete topology on the set X, every set is open; it follows that every
set is closed as well.

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