Page 215 - DMTH503_TOPOLOGY

Richa Nandra, Lovely Professional University Unit 25: The Nagata-Smirnov Metrization Theorem

Unit 25: The Nagata-Smirnov Metrization Theorem Notes

CONTENTS
Objectives
Introduction

25.1 The Nagata Smirnov Metrization Theorem
25.1.1 G Set

25.1.2 Nagata-Smirnov Metrization Theorem

25.2 Summary
25.3 Keywords
25.4 Review Questions
25.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define G set;

 State “The Nagata-Smirnov Metrization Theorem”;

 Understand the proof of “The Nagata Smirnov Metrization Theorem”.
Introduction

Although Urysohn solved the metrization problem for separable metric spaces in 1924, the
general metrization problem was not solved until 1950. Three mathematicians, J. Nagata, Yu. M.
Smirnov, and R.H. Bing, gave independent solutions to this problem. The characterizations of
Nagata and Smirnov are based on the existence of locally finite base, while that of Bing requires
a discrete base for the topology.
We will prove the regularity of X and the existence of a countably locally finite basis for X are
equivalent to metrizability of X.

25.1 The Nagata Smirnov Metrization Theorem

25.1.1 G Set


A subset A of a space X is called a G set in X if it equals the intersection of a countable collection

of open subsets of X.

Example 1: In a metric space X, each closed set is a G set- Given A  X, let U(A, ) denote

the  – neighbourhood of A. If A is closed, you can check that
A  U(A,1/ n)
n  Z 

LOVELY PROFESSIONAL UNIVERSITY 209