Page 186 - DMTH503_TOPOLOGY

Topology Richa Nandra, Lovely Professional University

Notes Unit 20: The Urysohn Metrization Theorem

CONTENTS
Objectives
Introduction

20.1 Metrization
20.2 Summary
20.3 Keywords

20.4 Review Questions
20.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Describe the Metrization;

 Explain the Urysohn Metrization Theorem;
 Solve the problems on Metrizability;
 Solve the problems on Urysohn Metrization Theorem.

Introduction

With Urysohn’s lemma, we now want to prove a theorem regarding the metrizability of
topological space. The idea of this proof is to construct a sequence of functions using Urysohn’s
lemma, then use these functions as component functions to embed our topological space in the
metrizable space.

20.1 Metrization

Given any topological space (X, T), if it is possible to find a metric  on X which induces the
topology T i.e. the open sets determined by the metric  are precisely the members of T, then X
is said to the metrizable.

Example 1: The set with usual topology is metrizable. For the usual metric on
induces the usual topology on . Similarly 2 with usual topology is metrizable.

Example 2: A discrete space (X, T) is metrizable. For the trivial metric induces the discrete
topology T on X.

Example 3: Prove that if a set is metrizable, then it is metrizable in an infinite number of
different ways.
Solution: Let X be a metrizable space with metric d.

Then  a metric d on X which defines a topology T on X.

180 LOVELY PROFESSIONAL UNIVERSITY