TITLE: Terminology for metric space with "anti-symmetric" distance
QUESTION [6 upvotes]: I'm interested in spaces that have a two-place function $d$ with non-negative real values, satisfying the following three conditions (for all $x$, $y$, $z$):

$d(x, x) = 0$
$d(x, y) + d(y, z) \geq d(x, z)$
If $d(x, y) = d(y, x) = 0$ then $x = y$

These conditions guarantee that the space is partially ordered with respect to the relation $d(x, y) = 0$, and a metric space with respect to the function $\max (d(x, y), d(y, x))$ (or alternatively $d(x, y) + d(y, x)$).
Do spaces like this have a standard name? Is there a good place to look for well-known results about them?
The function $d$ differs from a metric in that it need be neither symmetric nor positive definite. Wikipedia tells me that $d$ is similar to a "quasimetric", but the "anti-symmetry" condition 3 replaces the stronger condition that if $d(x, y) = 0$ then $x = y$.
Two examples of metric spaces that naturally arise from functions like $d$ are the Hausdorff metric on compact sets in a metric space and the metric on (equivalence classes of) measurable sets given by the measure of the symmetric difference.

REPLY [2 votes]: First observe that your 3 conditions are equivalent to requiring that $d:X\times X\to [0,+\infty)$ satisfies the following two conditions for all points $x,y,z\in X$:

$d(x,y)+d(y,z)\geq d(x,z)$,
$x=y \leftrightarrow d(x,y)=d(y,x)=0$.

In H.-P.A. Künzi, An introduction to quasi-uniform spaces they are called $T_0$-quasi-pseudometric. They have attracted attention recently thanks to the work of M. de Brecht (see Quasi-Polish spaces on arXiv) where they are called quasi-metric spaces. M. de Brecht showed that a large part of the theory of Polish spaces (separable completely metrisable spaces) can be extended to a certain class of quasi-metric spaces, namely the Quasi-Polish spaces.