Compactness of Metric Space

Sequential Compactness

Theorem

Let be a subset in . Then is closed and bounded iff every sequence in has a subsequence that converges to a point in .

Sequential Compactness

Let be a metric space. A set is called sequentially compact if every sequence in has a subsequence that converges to a point in .

Theorem

In a metric space, every sequentially compact set is bounded and closed.

The Converse is Not True in General

However, the converse is not true in general. For example, consider the metric space with the standard metric inherited from . Let a subset . Clearly is closed as a intersection of closed sets, and bounded. However, is not sequentially compact since a subsequence of any sequence of rationals converging to never converges to a point in .

Theorem

Let be a metric space. If is sequentially compact, then for any there is such that This is called a nearest point in to .

Proof By the definition of , there is a sequence in such that . Since is sequentially compact, has a convergent subsequence with . Then

Lemma Let be a metric space. If is sequentially compact, then for any there exists an integer and points in such that

Thrm Let be a metric space and let be sequentially compact. If is a family of open sets such that then there exists finitely many open sets from such that Proof

Covering and Compactness

Covering

Let be a metric space and . A collection of sets is called a covering of if . If in addition, each is open, then is called an open covering of .

Compactness

Let be a metric space. A set is called compact if for every open covering of there exists a finite subcollection of such that

Theorem

Let be a metric space. Then a set is compact iff it is sequentially compact.

Proof