Page 192 - DMTH503_TOPOLOGY

Topology Richa Nandra, Lovely Professional University

Notes Unit 21: The Tietze Extension Theorem

CONTENTS
Objectives
Introduction
21.1 Tietze Extension Theorem

21.2 Summary
21.3 Keywords
21.4 Review Questions
21.5 Further Readings

Objectives

After studying this unit, you will be able to:
 State the Tietze Extension Theorem;
 Understand the proof of Tietze Extension Theorem.

Introduction

One immediate consequence of the Urysohn lemma is the useful theorem called the Tietze
extension theorem. It deals with the problem of extending a continuous real-valued function
that is defined on a subspace of a space X to a continuous function defined on all of X. This
theorem is important in many of the applications of topology.

21.1 Tietze Extension Theorem

Suppose (X, T) is a topological space. The space X is normal iff every continuous real function of
defined point a closed subspace F of X into a closed interval [a, b] has a continuous extension.
f* : X  [a, b]
Proof:
(i) Suppose (X, T) is a topological space s.t. Every continuous real valued function f : F [a, b]
has a continuous extended function g : X [a, b] where F is a closed subset of X, [a, b] being
closed interval.
To prove X is a normal space.
Let F and F be two closed disjoint subsets of X.
1 2
Define a map f : F F [a, b]
1 2
s.t. f(x) = a if x  F and f(x) = b is x F .
1 2
This map f is certainly continuous over the subspace F F . By assumption, f can be
1 2
extended to a continuous map
g : X  [a, b] s.t.


 a if x F 1
g(x) = 

 b if x F 2
The map g satisfies Urysohn’s lemma and hence (X, T) is normal.

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