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.