Page 90 - DMTH503_TOPOLOGY

Topology Richa Nandra, Lovely Professional University

Notes Unit 9: The Metric Topology

CONTENTS
Objectives
Introduction

9.1 The Metric Topology
9.1.1 Metric Space
9.1.2 Pseudo Metric Space

9.1.3 Open and Closed Sphere
9.1.4 Boundary Set, Open Set, Limit Point and Closed Set
9.1.5 Convergence of a Sequence in a Metric Space
9.1.6 Theorems on Closed Sets and Open Sets
9.1.7 Interior, Closure and Boundary of a Point

9.1.8 Neighborhood
9.1.9 Theorems and Solved Examples
9.1.10 Uniform Convergence

9.2 Summary
9.3 Keywords
9.4 Review Questions
9.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define metric space and pseudo metric space;
 Understand the definitions of open and closed spheres, boundary set, open and closed set;
 Define convergence of a sequence in a metric space and interior, closure and boundary of
a point;
 Define neighborhood and limit point;
 Solve the problems on metric topology.

Introduction

The most important class of topological spaces is the class of metric spaces. Metric spaces provide
a rich source of examples in topology. But more than this, most of the applications of topology
to analysis are via metric spaces. The notion of metric space was introduced in 1906 by Maurice
Frechet and developed and named by Felix Hausdorff in 1914.
One of the most important and frequently used ways of imposing a topology on a set is to define
the topology in terms of a metric on the set. Topologies given in this way lie at the heart of

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