# Difference between revisions of "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 $x, y, z \in S$).

## Popular metrics

• The Euclidean metric on $\mathbb{R}^n$, with the "usual" meaning of distance