Let $X$ be a set. Let $d:X\times X \to \mathbb{R}$ be a function with the property that $d(x,y)\ge 0$ for all $x,y\in X$ . Then $d$ is a

1. semi-pseudometric if $d(x,y)=d(y,x)$ for all $x,y\in X$ ,
2. quasi-pseudometric if $d(x,z)\le d(x,y)+d(y,z)$ for all $x,y,z\in X$ .

$X$ equipped with a function $d$ described above is called a semi-pseudometric space or a quasi-pseudometric space, depending on whether $d$ is a semi-pseudometric or a quasi-pseudometric. A pseudometric is the same as a semi-pseudometric that is a quasi-pseudometric at the same time.

If $d$ satisfies the property that $d(x,y)=0$ implies $x=y$ , then $d$ is called a semi-metric if $d$ is a semi-pseudometric, or a quasi-metric if $d$ is a quasi-pseudometric.