Metric and metric space - Imported Knowlpedia Content

Let $X$ be a nonempty set. A function $d:X\times X\to\mathbb{R}$ is a metric on $X$ if for all $x,y,z\in X$ :

$d(x,y)\ge 0$ , and $d(x,y)=0$ if and only if $x=y$ .
$d(x,y)=d(y,x)$ .
$d(x,z)\le d(x,y)+d(y,z)$ (triangle inequality).

The pair $(X,d)$ is called a metric space.

Metrics allow one to define balls, open sets, and notions of convergence and completeness.

Examples

On $\mathbb{R}^n$ , the Euclidean metric $d(x,y)=\|x-y\|_2$ .
The discrete metric: $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ otherwise.
On a normed vector space $(X,\|\cdot\|)$ , $d(x,y)=\|x-y\|$ is a metric.