Metric space

A metric space is a pair, $(S, d)$ of a set $S$ and a metric $d: S \times S \to \mathbb{R}_{\geq 0}$ . The metric $d$ represents a distance function between pairs of points of $S$ , and we require it to obey symmetry ( $d(x, y) = d(y, x)$ ), positivity ( $d(x, y) \geq 0$ and $d(x, y) = 0$ if and only if $x = y$ ) and the triangle inequality ( $d(x, y) + d(y, z) \geq d(x, z)$ for all points $x, y, z \in S$ ).