metric space

The tuple $(X,d)$ is a metric space if $X$ is a non-empty set and $d:X\times X\to \mathbb{R}$ satisfies:

• (non-negativity) $d(x,y)\ge 0$ for all $x,y$;
• (symmetry) $d(x,y)=d(y,x)$ for all $x,y$;
• (triangle inequality) $d(x,y)\le d(x,z)+d(z,y)$.