metric

Definition

A metric is a distance function $d : X \times X \to [0,\infty)$ for set $X$ satisfying axioms

• triangle inequality, $d(x,y) + d(y,z) \geq d(x,z)$
• point inequality, $0 \geq d(x,x)$
  • so $0 = d(x,x)$
• separation, $x=y$ if $d(x,y) = 0$
  • so $x=y$ iff $d(x,y)=0$
• symmetry $d(x,y) = d(y,x)$

References

1. https://ncatlab.org/nlab/show/metric+space