Page 223 - DMTH503_TOPOLOGY

Sachin Kaushal, Lovely Professional University Unit 27: Complete Metric Spaces

Unit 27: Complete Metric Spaces Notes

CONTENTS
Objectives
Introduction

27.1 Cauchy’s Sequence
27.2 Complete Metric Space
27.3 Theorems and Solved Examples

27.4 Summary
27.5 Keywords
27.6 Review Questions
27.7 Further Readings

Objectives

After studying this unit, you will be able to:

 Define Cauchy’s sequence;
 Solve the problems on Cauchy’s sequence;
 Define complete metric space;
 Solve the problems on complete metric spaces.

Introduction

The concept of completeness for a metric space is basic for all aspects of analysis. Although
completeness is a metric property rather than a topological one, there are a number of theorems
involving complete metric spaces that are topological in character. In this unit, we shall study
the most important examples of complete metric spaces and shall prove some of these problems.

27.1 Cauchy’s Sequence

A sequence <x > in a metric space X is said to be a Cauchy sequence in X if given > 0 there exists
n
a positive integer n such that
o
d(x , x ) < where m, nn .
m n o
Alternative definition: A sequence <x > is Cauchy if given> 0, there exists a positive integer
n
n such that
o
d(x , x ) < for all nn and for all p1.
n+p n o
Theorem 1: Every convergent sequence in a metric space is a Cauchy sequence.

Proof: Let (X, d) be a metric space.
Let <x > be a convergent sequence in X.
n
Suppose lt x = x.
n®¥ n

LOVELY PROFESSIONAL UNIVERSITY 217