Metric

A metric on a set $X$ is a function $d\colon X\times X\to[0,\infty)$ such that for all $x,y,z\in X$:

1. (Identity of indiscernibles) $d(x,y)=0$ if and only if $x=y$.
2. (Symmetry) $d(x,y)=d(y,x)$.
3. (Triangle inequality) $d(x,z)\le d(x,y)+d(y,z)$.

A metric is the basic structure underlying a metric space ; it determines open balls and hence the metric-induced topology .

Examples:

• On $\mathbb{R}^n$, the Euclidean metric $d(x,y)=\|x-y\|_2$.
• On any set $X$, the discrete metric $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ otherwise.