generalization of a pseudometric

Let $X$ be a set. Let $d:X\times X\to ℝ$ 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.