Page 170 - DMTH503_TOPOLOGY

Topology Sachin Kaushal, Lovely Professional University

Notes Unit 18: Normal Spaces, Regular Spaces and
Completely Regular Spaces

CONTENTS

Objectives
Introduction
18.1 Normal Space

18.2 Regular Space
18.3 Completely Regular Space
18.4 Summary
18.5 Keywords
18.6 Review Questions

18.7 Further Readings

Objectives

After studying this unit, you will be able to:
 Define normal space;

 Solve the problems on normal space;
 Discuss the regular space;
 Describe the completely regular space;
 Solve the problems on regular and completely regular space.

Introduction

Now we turn to a more through study of spaces satisfying the normality axiom. In one sense, the
term “normal” is something of a misnomer, for normal spaces are not as well-behaved as one
might wish. On the other hand, most of the spaces with which we are familiar do satisfy this
axiom, as we shall see. Its importance comes from the fact that the results one can prove under
the hypothesis of normality are central to much of topology. The Urysohn metrization theorem
and the Tietze extension theorem are two such results; we shall deal with them later. We shall
study about regular spaces and completely regular spaces.

18.1 Normal Space

A topological space (X, T) is said to be normal space if given a pair of disjoint closed sets C ,
1
C  X.
2
 disjoint open sets G , G  X s.t. C  G , C  G .
1 2 1 1 2 2
Example 1: Metric spaces are normal.

Solution: Before proving this, we need a preliminary fact. Let X be a metric space with metric d.
Given a subset A  X define the distance d(x, A) from a point x  X to A to the greatest lower

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