Metric and metric space

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

1. $d(x,y)\ge 0$, and $d(x,y)=0$ if and only if $x=y$.
2. $d(x,y)=d(y,x)$.
3. $d(x,z)\le d(x,y)+d(y,z)$ (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.