# example of an “almost-metric” without symmetry

It is not difficult to find an "almost-metric" $$d$$ that satisfies all axioms of a metric except the triangle inequality. It should also be possible to construct a function $$d$$ that satisfies all axioms except the symmetry, but I was unable to find one. (But I'm sure there must be one.)

Is there an example of a "almost-metric" that is not symmetric but satisfies the other axioms of a metric (positive-definiteness, triangle inequality)?

To be specific (because $$d$$ should not be symmetric) we want

$$d(a,b) + d(b,c) \geq d(a,c)$$

to hold. It is certainly interesting to find an example in $$\mathbb R^n$$ (even if just for a specific $$n$$) but I'm also interested in other "almost-metric-spaces".

I just found out that these $$d$$ are called Quasimetrics. An example on $$\mathbb R$$ is the following:
$$d(x,y) = \begin{cases} x - y & x \geq y \\ 1 & \text{otherwise}\end{cases}$$