Page 144 - DMTH503_TOPOLOGY

Topology Sachin Kaushal, Lovely Professional University

Notes Unit 15: Local Compactness

CONTENTS
Objectives
Introduction

15.1 Locally Compact
15.2 Summary
15.3 Keywords

15.4 Review Questions
15.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Describe the local compactness;

 Solve the problems on local compactness;
 Explain the theorems on local compactness.

Introduction

In this unit, we study the notion of local compactness and we prove the theorems that every
continuous image of a locally compact space is locally compact and many other theorems.

15.1 Locally Compact

Let (X, T) be a topological space and let x  X be arbitrary. Then X is said to be locally compact
at x if the closure of any neighborhood of x is compact.
X is called locally compact if it is compact at each of its points, but need not be compact as whole.
Alternative definition: A topological space (X, T) is locally compact if each element x  X has a
compact neighborhood.

Example 1: Show that is locally compact.
Solution: Let x  R be arbitrary.
Evidently, S (x) = S [x]
r
r
S [x] is compact, being closed and bounded subset of . Thus the closure of the neighborhood
r
S (x) of x is compact and hence the result.
r
Example 2: Show that compactness  locally compact.
Solution: Let (X, T) be a compact topological space. To prove that X is locally compact.

For this, we must show that the closure of any neighborhood of any point x  X is compact. This
follow from the fact that X is the neighborhood of each of its points and X  X, X is compact.

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