Page 96 - DMTH401_REAL ANALYSIS

Real Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 8: Completeness and Compactness

CONTENTS
Objectives
Introduction
8.1 Completeness and Compactness
8.2 Cantor's Theorem
8.3 Perfect Set
8.3.1 Perfect Sets are Uncountable
8.4 Cantor Middle Third Set
8.5 Baire Category Theorem
8.6 Compactness and Cantor Set
8.7 Summary
8.8 Keywords
8.9 Review Questions
8.10 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss Completeness and Compactness
 Describe the Cantor's theorem
 Explain Baire category theorem
 Describe Compactness and Cantor set

Introduction

In earlier unit you have studied about the compactness and connectedness of the set. As you all
come to know about the Contraction Mapping Theorem. After understanding the concept of
Total boundedness let us go through the explanation of completeness and connectedness.

8.1 Completeness and Compactness

Theorem: Subspace C of complete metric M compact iff closed and totally bounded.
Proof: () C closed, totally bounded since "  > 0 open cover B(x, ) (x  C) has finite subcover.
() Every sequence in C has Cauchy subsequence, converges to point of M since M complete. C
closed so limit in C.
Lemma: If M subspace of N totally bounded so is M.


Proof: Fix  > 0. Let F cM be finite s.t. M  B(x, ). Then
xF 2
æ  ö
M   B x, ÷  B(x, )

ç
x F è 2 ø

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