Page 73 - DMTH401_REAL ANALYSIS

Sachin Kaushal, Lovely Professional University Unit 4: Compactness

Unit 4: Compactness Notes

CONTENTS
Objectives
Introduction
4.1 Compactness
4.2 Compactness of Subsets

4.3 Intersections of Closed Sets
4.4 Compactness of Products
4.5 Compactness and Continuity
4.6 Compact Sets in  n
4.7 Sequential Compactness
4.8 Summary
4.9 Keywords

4.10 Review Questions
4.11 Further Readings
Objectives

After studying this unit, you will be able to:
 Discuss the compactness of a set
 Explain intersection of closed set
 Discuss compactness and continuity
 Describe sequential compactness

Introduction

In last unit you have studied about matric spaces. You all go through concept of open sets, limit
points of sets in last unit. This unit will provide you explanations of compactness of a set.

4.1 Compactness

Definition:

 A cover of A is a collection U of sets whose union contains A.
 A subcover is a subcollection of U which still covers A.
 A subcover is open if its members are all open.
Definition: Topological space T is compact if every open cover has finite subcover.

Theorem: (Heine-Borel). Any closed bounded interval [a, b]  is compact.
Proof: Let U be open cover of [a, b]. Let
A = {x  [a, b] : [a, x] covered by finite subfamily of U}

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