MathItems

A set $X$, whose elements we shall call points, is said to be a metric space if with any two points $p$ and $q$ of $X$ there is associated a real number $d(p,q)$, called the distance from $p$ to $q$, such that

1. $d(p,q)>0$ if $p\neq0$; $d(p,p)=0$;
2. $d(p,q)=d(q,p)$;
3. $d(p,q)\leq d(p,r)+d(r,q)$, for any $r\in X$.

Any function with these three properties is called a distance function, or a metric.