metric space axioms

The metric space axioms are an axiomatic system for distance that is agnostic to interpretation.

Formally,

1. the distance from a point to itself is zero
2. the distance between two distinct points is positive
3. the distance from $x$ to $y$ is equal to the distance from $y$ to $x$ (symmetric)
4. the distance between two points is less than or equal to the sum of distances to a third point (the triangle inequality)