Open and Closed Sets

Bounded Subset

A subset of is bounded if there exist and such that .

Open and Closed Subset

A subset of is open in if for every there exists such that . A subset of is closed if is open.

e.g.

• In open intervals are open and closed intervals are closed.
• In any metric space both and are both open and closed.
• In a metric space with the discrete metric every point is open.

Theorem

In a metric space , a subset is open iff iff every has an open ball such that .

Lemma Open balls are open. Proof Suppose is a open ball on metric space . For all , we have . Let , consider ball . For all , we have Thus , it follows that , hence is open.

Theorem

Let be a metric space. For any set , and are open.

Proof We have Since and is open, is also open.

Lemma The intersection of finitely many open sets is open, that is if are open in then is open in .

Proof For all , since are open, exist such that . Let , then for all , we have Hence for all . Therefore for all and thus .

Lemma The union of arbitrarily many open sets is open.

Corollary

The union of finitely many closed sets is closed, that is if are closed in then is closed in . And the intersection of arbitrarily many closed sets is closed.

Proof We observe that Therefore by applying the above lemma we have is closed.

Structure of Open Sets in

Every open sets in is a countable union of disjoint open intervals.

Proof Let be open in . For each we can find such that . Let Then . One can show, for any with , either or . This completes the proof.

Thrm In a metric space a set is open iff is closed.

Thrm Let be a metric space and . Then Proof .

Thrm Let be a metric space and . Then and are closed.

Proof Recall that and is open. Thus is closed. Recall also that and both and are open. Hence is closed.

Thrm is the largest open set contained in . is the smallest closed set containing .

Proof If is an open set contained in , then . If is a closed set containing , then and hence .