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