Page 49 - DMTH401_REAL ANALYSIS

Sachin Kaushal, Lovely Professional University Unit 3: Matric Spaces

Unit 3: Matric Spaces Notes

CONTENTS
Objectives
Introduction
3.1 Matric Spaces
3.1.1 Space Properties

3.1.2 Distance between Points and Sets; Hausdorff Distance and Gromov Metric
3.1.3 Product Metric Spaces
3.2 Modulus of Real Number
3.2.1 Properties of the Modulus of Real Number
3.3 Neighbourhoods
3.4 Open Sets
3.5 Limit Point of a Set

3.5.1 Bulzano Weierstrass Theorem
3.6 Closed Sets
3.7 Compact Sets
3.8 Summary
3.9 Keywords
3.10 Review Questions
3.11 Further Readings

Objectives

After studying this unit, you will be able to:

 Define the modulus of a real number
 Describe the notion of a neighbourhood of a point on the line
 Define an open set and give examples
 Discuss the limit points of a set
 Define a closed set and establish its relation with an open set

 Explain the meaning of an open covering of a subset of real numbers
Introduction

You all are quite familiar with an elastic string or a rubber tube or a spring. Suppose you have
an elastic string. If you first stretch it and then release the pressure, then the string will come
back to its original length. This is a physical phenomenon but in Mathematics, we interpret it
differently. According to Geometry, the unstreched string and the stretched string are different
since there is a change in the length. But you will be surprised to know that according to another
branch of Mathematics, the two positions of the string are identical and there is no change. This
branch is known as Topology, one of the most exciting areas of Mathematics.

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