A pseudometric space is a set $X$ together with a non-negative real-valued function $d:X\times X\longrightarrow\mathbb{R}$ (called a pseudometric) such that, for every $x,y,z\in X$,

• $d(x,x)=0$.

• $d(x,y)=d(y,x)$

• $d(x,z)\leq d(x,y)+d(y,z)$

In other words, a pseudometric space is a generalization of a metric space in which we allow the possibility that $d(x,y)=0$ for distinct values of $x$ and $y$.