Metric (analysis)

A metric $d$ on a set $S$ is a function $d: S \times S \to \mathbb{R}$ which obeys the following three properties:

  • Symmetry: $d(x, y) = d(y, x)$ for all points $x, y \in S$.
  • Positivity: $d(x, y) \geq 0$ for all $x, y \in S$ and $d(x, y) = 0$ if and only if $x = y$.
  • The triangle inequality: $d(x, y) + d(y, z) \geq d(x, z)$ for all $x, y, z \in S$.

Together, the set $S$ and the metric $d$ form a metric space.

Every metric space can be used to form a topology by considering taking the set of open balls as a topological basis (i.e. the sets $B_{x,\epsilon}:=\{y\in S:d(x,y)<\epsilon\}$).

Common metrics

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