Quasi-metric

Let $\mathbb X$ be a nonempty set. A function $d:\mathbb{X}\times\mathbb{X}\to[0,\infty)$ which satisfies the following conditions for all $x,y\in\mathbb X$

1) $d(x,y)=0$ if and only if $x = y$ (the identity axiom);

2) $d(x,y) + \rho(y,z) \geq d(x,z)$ (the triangle axiom);

is called quasi-metric. A pair $(\mathbb X, d)$ is a quasi-metric space.

The difference between a metric and a quasi-metric is that a quasi-metric does not possess the symmetry axiom (in the case we allow $d(x,y)\ne d(y,x)$ for some $x,y\in \mathbb X$ ).

Reference

[Sch]	V. Schroeder, "Quasi-metric and metric spaces". Conform. Geom. Dyn. 10, 355 - 360 (2006) Zbl 1113.54014
[Wil]	W. A. Wilson, "On Quasi-Metric Spaces". American Journal of Mathematics Vol. 53, No. 3 (1931), pp. 675-684 Zbl 0002.05503