Metric/Metric space/Definition

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

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

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