Page 232 - DMTH503_TOPOLOGY

Topology Sachin Kaushal, Lovely Professional University

Notes Unit 28: Compactness in Metric Spaces

CONTENTS
Objectives
Introduction

28.1 Bolzano Weierstrass Theorem
28.1.1 Sequentially Compact
28.1.2 Lebesgue Number

28.1.3 Totally Bounded Set
28.1.4 Compactness in Metric Spaces
28.2 Theorems and Solved Examples
28.3 Summary
28.4 Keywords

28.5 Review Questions
28.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Know the Bolzano Weierstrass theorem and BWP;
 Define sequentially compact and lebesgul measure;
 Define totally bounded set;

 Describe the compactness in metric spaces;
 Solve the related problems.

Introduction

We have already shown that compactness, limit point compactness and sequentially compact
are equivalent for metric spaces. There is still another formulation of compactness for metric
spaces, one that involves the notion of completeness. We study it in this unit. As an application,
we shall prove a theorem characterizing those subspaces of (X, R ), that are compact in the
n
uniform topology.
28.1 Bolzano Weierstrass Theorem

A closed and bounded infinite subset of R contains a limit point.
Bolzano Weierstrass Property: A metric space (X, d) is said to have the Bolzano weierstrass
property if every infinite subset of X has a limit point.
In brief, ‘Bolzano Weierstrass Property’ is written as B.W.P. A space with B.W.P. is also called
Frechet compact space.

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