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Richa Nandra, Lovely Professional University Unit 19: The Urysohn Lemma

Unit 19: The Urysohn Lemma Notes

CONTENTS
Objectives
Introduction

19.1 Urysohn’s Lemma
19.1.1 Proof of Urysohn’s Lemma
19.1.2 Solved Examples

19.2 Summary
19.3 Keywords
19.4 Review Questions
19.5 Further Readings

Objectives

After studying this unit, you will be able to:

 State Urysohn’s lemma;
 Understand the proof of Urysohn’s lemma;

 Solve the problems on Urysohn’s lemma.
Introduction

Saying that a space X is normal turns out to be a very strong assumption. In particular, normal
spaces admit a lot of continuous functions. Urysohn’s lemma is sometimes called “the first
non-trivial fact of point set topology” and is commonly used to construct continuous functions
with various properties on normal spaces. It is widely applicable since all metric spaces and all
compact Hausdorff spaces are normal. The lemma is generalized by (and usually used in the
proof of) the Tietze Extension Theorem.

19.1 Urysohn’s Lemma

In topology, Urysohn’s lemma is a lemma that states that a topological space is normal iff any
two disjoint closed subsets can be separated by a function.
This lemma is named after the mathematician Pavel Samuilovich Urysohn.

19.1.1 Proof of Urysohn’s Lemma

Urysohn’s Lemma: Consider the set R with usual topology where R = {x  R : 0  x  1}

A topological space (X, T) is normal iff given a pair of disjoint closed sets A, B  X, there is a
continuous functions.
f : X  R s.t. f(A) = {0} and f(B) = {1}.

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