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 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).