Metric/Metric space/Definition

Metric space

Let ${}M$ be a set. A mapping ${}d\colon M\times M\rightarrow \mathbb {R}$ is called a metric (or a distance function), if for all ${}x,y,z\in M$ the following conditions hold:

1. ${}d{\left(x,y\right)}=0$ if and only if ${}x=y$ (positivity),
2. ${}d{\left(x,y\right)}=d{\left(y,x\right)}$ (symmetry), and
3. ${}d{\left(x,y\right)}\leq d{\left(x,z\right)}+d{\left(z,y\right)}$ (triangle inequality).

A metric space is a pair ${}(M,d)$, where ${}M$ is a set and ${}d\colon M\times M\rightarrow \mathbb {R}$ is a metric.