A metric on a Set $X$ is a map $d: X \times X \rightarrow \mathbb{R}$ (i.e. from the Cartesian square of $X$ to the Real numbers), that satisfies the conditions:

• symmetry: $d(x,y) = d(y,x)$
• positivity: $d(x,y) \geq 0$, and $=0$ if, and only if, $x=y$.
• Triangle inequality: $d(x,y) \leq d(x,z) + d(z,y)$.