Page 254 - DMTH503_TOPOLOGY

Topology Richa Nandra, Lovely Professional University

Notes Unit 31: Baire Spaces

CONTENTS
Objectives
Introduction
31.1 Baire Spaces
31.1.1 Definition – Baire Space

31.1.2 Baire’s Category Theory
31.1.3 Baire Category Theorem
31.2 Summary
31.3 Keywords
31.4 Review Questions
31.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Know about the Baire spaces;

 Understand the Baire’s category theory;
 Understand the Baire’s category theorem.

Introduction

In this unit, we introduce a class of topological spaces called the Baire spaces. The defining
condition for a Baire space is a bit complicated to state, but it is often useful in the applications,
in both analysis and topology. Most of the spaces we have been studying are Baire spaces. For
instance, a Hausdorff space is a Baire space if it is compact, or ever locally compact. And a
metrizable space X is a Baire space if it is topologically complete, that is, if there is a metric for
X relative to which X is complete.
Then we shall give some applications, which ever if they do not make the Baire condition seem
any more natural, will at least show what a useful tool it can be in feet, it turns out to be a very
useful and fairly sophisticated tool in both analysis and topology.
31.1 Baire Spaces

31.1.1 Definition – Baire Space

A space X is said to be a Baire space of the following condition holds. Given any countable
collection {A } of closed sets of X each of which has empty interior in X, their unionA also has
n n
empty interior in X.

Example 1: The space  of rationals is not a Baire space. For each one-point set in 
is closed and has empty interior in ; and  is the countable union of its one-point subsets.
The space Z , on the other hand, does form a Baire space. Every subset of Z is open, so that there
+ +

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