Definition

metric

The function $d: S \times S \to \mathbb{R}_{+}$ is a metric if

1. nonnegativity: for all $x, y \in S$ have $d(x,y) \geq 0$,
2. zero-equality: for all $x, y \in S$ have $d(x,y) = 0$ if and only if $x = y$,
3. symmetry: for all $x, y \in S$ have $d(x,y) = d(y,x)$,
4. triangle inequality: for all $x, y, z \in S$ have $d(x,z) \leq d(x,y) + d(y,z)$.