\begin{definition}[Definition:Quasimetric]
A '''quasimetric''' on a set $X$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions:
{{begin-axiom}}
{{axiom | n = \text M 1
        | q = \forall x \in A
        | m = \map d {x, x} = 0
}}
{{axiom | n = \text M 2
        | q = \forall x, y, z \in A
        | m = \map d {x, y} + \map d {y, z} \ge \map d {x, z}
}}
{{axiom | n = \text M 4
        | q = \forall x, y \in A
        | m = x \ne y \implies \map d {x, y} > 0
}}
{{end-axiom}}
Note the numbering system of these conditions. They are numbered this way so as to retain consistency with the metric space axioms, of which these are a subset.
The difference between a '''quasimetric''' and a metric is that a '''quasimetric''' does not insist that the distance function between distinct elements is commutative, that is, that $\map d {x, y} = \map d {y, x}$.
\end{definition}