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Sachin Kaushal, Lovely Professional University Unit 23: The Stone–Cech Compactification

Unit 23: The Stone-Cech Compactification Notes

CONTENTS
Objectives
Introduction

23.1 Compactification
23.1.1 One Point Compactification
23.1.2 Stone-Cech Compactification

23.2 Summary
23.3 Keywords
23.4 Review Questions
23.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Describe the compactification;
 Define the Stone-Cech compactification;
 Explain the related theorems.

Introduction

We have already studied one way of compactifying a topological space X, the one-point
compactification; it is in some sense the minimal compactification of X. The Stone-Cech
compactification of X, which we study now, is in some sense the maximal compactification of X.
It was constructed by M. Stone and E. Cech, independently, in 1937. It has a number of applications
in modern analysis. The Stone-Cech compactification is defined for all Tychonoff Spaces and has
an important extension property.

23.1 Compactification

A compactification of a space X is a compact Hausdorff space Y containing X as a subspace such
that X = Y. Two compactifications Y, and Y of X are said to be equivalent if there is a
2
homeomorphism h : Y Y such that h(x) = x for every xX.
1 2
Remark: If X has a compactification Y, then X must be completely regular, being a subspace of
completely regular space Y. Conversely, if X is completely regular, then X has a compactification.

Lemma 1: Let X be a space; suppose that h : XX is an imbedding of X in the compact Hausdorff
space Z. Then there exists a corresponding compactification Y of X; it has the property that there
is an imbedding H : YZ that equals h on X. The compactification Y is uniquely determined up
to equivalence.
We call Y the compactification induced by the imbedding h.

Proof: Given h, let X denote the subspace h(X) of Z, and let Y denote its closure of Z.
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