Metric

Definition

Let $X$ be a set.

A metric on $X$ is a function

$$d:X\times X\to \mathbb{R}$$

that assigns to every pair $(x,y)\in X\times X$ a real number $d(x,y)$, such that for all $x,y,z\in X$:

Axioms

Non-negativity

$$d(x,y)\ge 0$$

Identity of indiscernibles

$$d(x,y)=0⟺x=y$$

Symmetry

$$d(x,y)=d(y,x)$$

Triangle inequality

$$d(x,z)\le d(x,y)+d(y,z)$$