Hemimetric space

In mathematics, a hemimetric space is a generalization of a metric space, obtained by removing the requirements of identity of indiscernibles and of symmetry. It is thus a generalization of both a quasimetric space and a pseudometric space, while being a special case of a prametric space.

Definition

A hemimetric on a set $X$ is a function $d\colon X\times X\to \mathbb {R}$ such that

1. $\,\!d(x,y)\geq 0$ (positivity);
2. $\,\!d(x,z)\leq d(x,y)+d(y,z)$ (subadditivity/triangle inequality);
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 (the identity of indiscernibles).

A symmetric hemimetric is a pseudometric.

A hemimetric that can discern points is a quasimetric.

A hemimetric induces a topology on $X$ in the same way that a metric does, a basis of open sets being

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

where $B_{r}(x)=\{y\in X:d(x,y)<r\}$ is the r-ball centered at $x$.

References

• hemimetric at PlanetMath.