Definition (Metric)

The function $d:\mathscr{X}\times\mathscr{X}\to[0,\infty)$ is called a distance on a metric (or simply a metric) if it satisfies $\forall x,y,z\in\mathscr{X}$

1. Non-negativity: $d(x,y)\ge0$
2. Reflexive: $d(x,y)=0 \iff x=y$
3. Symmetry: $d(x,y)=d(y,x)$
4. Sub-additivity or Triangular inequality: $d(x,z)+d(z,y)\ge d(x,y)$