# Metric Spaces¶

A surface is more than just a set of points. Points on a surface have a notion
of *closeness* that doesn’t exist with a set unless we add some structure.

One way we can introduce the idea of closeness is to introduce the idea of the distance between points. That is, a function that gives us a number for any pair of points.

To be a **distance function**, our function must meet some additional
requirements:

- all distances must be non-negative:
- the distance between and is zero if and only if and are the same point:
- the distance between and is the same as the distance between and . In other words, the distance function is always
*symmetric*: - finally, the distance between two points can never exceed the sum of the distance between each of the points and a third point. This is often referred to as the
*triangle rule*:

A distance function is often called a **metric**. A set of points with a
distance function is called a **metric space**.

A metric space clearly has a notion of closeness. A point is closer to than is if .