A **metric space** is a nonempty set along with a function such that

- , with equality iff .
- .

Given a point and a positive number , the **open ball** centered at with radius is

and the **closed ball** centered at with radius is

A set is said to be **open** if for each , there exists such that . A set is said to be **closed** if is open.

Theorem 1An open ball is an open set.

*Proof:* Let and , and let . Put . Then if ,

so that is open.

Theorem 2If is a metric space, then and are open, the union of any collection of open sets is open, and the intersection of finitely many closed sets is open.

*Proof:* It is clear that is open, as there is no . Since for any , is open.

Let be a collection of open sets in . Put . If , then for some . Since is open, there exists such that . Hence is open.

Let be open sets in . If then there exist such that and . Put , then , so that is closed. The general result follows from induction.

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