hemimetric

A hemimetric on a set $X$ is a function$d:X\times X\to ℝ$ such that

1. $d(x,y)\ge 0$;

2. $d(x,z)\le d(x,y)+d(y,z)$;

3. $d(x,x)=0$;

for all $x,y,z\in X$.

Hence, essentially $d$ is a metric which fails to satisfy symmetry and the property that distinct points have positive distance.A hemimetric induces a topology on $X$ in the same way that a metric does, a basis of open sets being

$$\{B(x,r):x\in X,r>0\},$$

where is the $r$-ball centered at $x$.