A set of points together with a function, d, called a metric function or distance function. The function assigns a positive real number to each pair of points, called the distance between them, such that:
2. For any two distinct points x and y, d(x,y)>0;
3. For any two points x and y, not necessarily distinct,
d(x,y) = d(y,x).
4. For any three points x, y, and z, that are not necessarily distinct,
d(x,z) <= d(x,y) + d(y,z).
The distance from x to z does not exceed the sum of the distances from x to y and from y to z. The sum of the lengths of two sides of a triangle is equal to or exceeds the length of the third side.