**Definition.** A metric on a set is a function: such that

1) iff ;

2) for all ;

3) for all .

A set equipped with a metric is called a metric space.

**Examples.** 1) The Euclidean distance is a metric on .

2) and are metrics on the space of continuous functions on .

3) If is a metric on and , then is a metric on .

4) If and are metric spaces, the product metric on is given by

.

5) Some other metrics used on :

, .

**Definition.** Let be a metric space. If and , the (open) ball of radius about is .

A set is open if for every , such that , and closed if the complement is open.

**Remark. ** and are both open and closed. The union of any family of open sets is open, and the intersection of any family of closed sets is closed. The intersection (resp. union) of any finite family of open (resp. closed) sets is open (resp. closed). Indeed, if are open and , for each , $\exists r_j>0$ such that , and then where , so is open.

**Definition.** If , then the interior of , denoted by the union of all open sets is the largest open set contained in , which is called the interior of and is denoted by . The intersection of all closed sets is the smallest closed set containing ; it is called the closure of and is denoted by . is said to be dense in if , and nowhere dense if has empty interior. is called separable if it has a countable dense subset. (For example, is a countable dense subset of .) A sequence in converges to (symbolically: or ) if .

**Proposition. **If is a metric space, , and , the following are equivalent:

1) .

2) for all .

3) There is a sequence in that converges to .

Proof. If , then is a closed set containing but not , so . Conversely, if $x\not\in\bar{E}$, since is open $\exists r>0$ such that . Thus 1) is equivalent to 2). If 2) holds, for each , $\exists x_n\in B(n^{-1}, x)\cap E$, so that . On the other hand, if , then for all , so no sequence of can converge to . Thus 2) is equivalent to 3).

**Remark.** If and are metric spaces, a map is called continuous at if for every , such that whenever . In other words, . The map is called continuous if it is continuous at each and uniformly continuous if the in the definition of continuity can be chosen independent of .

**Proposition.** is continuous iff is open in for every open .

Proof. If the latter condition holds, then for every and , the set is open and contains , so it contains some ball about ; this means that is continuous at . Conversely, suppose that is continuous and is open in . For each , such that , and for each , such that . Thus is open. $\Box$

“I understand now that the vulnerability I’ve always felt is the greatest strength a person can have. You can’t experience life without feeling life. What I’ve learned is that being vulnerable to somebody you love is not a weakness, it is strength.” ~ Elisabeth Shue

References:

[1] Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2ed, page 13-14.

[2] Purpose Fairy’s 21-Day Happiness Challenge, http://www.jrmstart.com/wordpress/wp-content/uploads/2014/10/Free+eBook+-+PurposeFairys+21-Day+Happiness+Challenge.pdf.