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 *d* 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: d(x,y)>=0
- the distance between
*x*and*y*is zero if and only if*x*and*y*are the same point: d(x,y)=0 iff x=y - the distance between
*x*and*y*is the same as the distance between*y*and*x*. In other words, the distance function is always*symmetric*: d(x,y)=d(y,x) - 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*: d(x,z)<=d(x,y)+d(y,z)

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 *y* is closer to *x*
than *z* is if d(y,x)<d(z,x).

**UPDATE**: next post

Tweet

The original post was in the category: poincare_project but I'm still in the process of migrating categories over.