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Sachin Kaushal, Lovely Professional University Unit 24: Local Finiteness and Paracompactness

Unit 24: Local Finiteness and Paracompactness Notes

CONTENTS
Objectives
Introduction

24.1 Local Finiteness
24.1.1 Countably Locally Finite
24.1.2 Open Refinement and Closed Refinement

24.2 Paracompactness
24.3 Summary
24.4 Keywords
24.5 Review Questions
24.6 Further Readings

Objectives

After studying this unit, you will be able to:
 Define local finiteness and solve problems on it;

 Define countably locally finite, open refinement and closed refinement;
 Understand the paracompactness and theorems on it.
Introduction

In this unit we prove some elementary properties of locally finite collections and a crucial
lemma about metrizable spaces.

The concept of paracompactness is one of the most useful generalization of compactness that has
been discovered in recent years. It is particularly useful for applications in topology and
differential geometry. Many of the spaces that are familiar to us already are paracompact. For
instance, every compact space is paracompact; this will be an immediate consequence of the
definition. It is also true that every metrizable space is paracompact; this is a theorem due to
A.H. Stone, which we shall prove. Thus the class of paracompact space includes the two most
important classes of spaces we have studied. It includes many other spaces as well.

24.1 Local Finiteness

Definition: Let X be a topological space. A collection  of subsets of X is said to be a locally finite
in X if every point of X has a neighbourhood that intersects only finitely many elements of .

Example 1: The collection of intervals
 = {(n, n + 2)| n  )}

is locally finite in the topological space , on the other hand, the collection
 = {0, 1/n} | n  }
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