Let $X$ be a set, and let $d:X\times X\to\mathbb{R}$. We say that $d$ is a metric on $X$ if for every $x,y,z\in X$ the following conditions hold.

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

If $d$ is a metric on $X$, then the pair $(X,d)$ is called a metric space.