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Richa Nandra, Lovely Professional University Unit 14: Limit Point Compactness

Unit 14: Limit Point Compactness Notes

CONTENTS
Objectives
Introduction

14.1 Limit Point Compactness and Sequentially Compact
14.1.1 Limit Point Compactness
14.1.2 Sequentially Compact

14.2 Summary
14.3 Keywords
14.4 Review Questions
14.5 Further Readings

Objectives

After studying this unit, you will be able to:

 Define limit-point compactness and solve related problems;
 Define the term sequentially compact and solve questions on it.

Introduction

In this unit, we introduce limit point compactness. In some ways, this property is more natural
and intuitive than that of compactness. In the early days of topology, it was given the name
“compactness”, while the open covering formulation was called “bicompactness”. Later, the
word “compact” was shifted to apply the open covering definition, leaving this one to search for
a new name. It still has not found a name on which everyone agrees. On historical grounds,
some call it “Frechet compactness” others call it the “Bolzano–Weierstrass property”. We have
invented the term “limit point compactness”. It seems as good a term as any at least it describes
what the property is about.

14.1 Limit Point Compactness and Sequentially Compact

14.1.1 Limit Point Compactness

A space X is said to be limit point compact if every infinite subset of X has a limit point.
Theorem 1: Compactness implies limit point compactness, but not conversely.
Proof: Let X be a compact space. Given a subset A of X, we wish to prove that if A is infinite, then
A has a limit point. We prove the contra positive – if A has no limit point, then A must be finite.
So suppose A has no limit point. Then A contains all its limit points, so that A is closed. Further
more, for each a A, we can choose a neighborhood U of a such that U intersects A in the point
a a
a alone. The space of X is covered by the open set X – A and the open sets U ; being compact, it can
a
be covered by finitely many of these sets. Since X – A does not intersect A, and each set U
a
contains only one point of A, the set A must be finite.

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