\begin{definition}[Definition:Pseudometric]
A '''pseudometric''' on a set $A$ 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 3
        | q = \forall x, y \in A
        | m = \map d {x, y} = \map d {y, x}
}}
{{end-axiom}}
The difference between a '''pseudometric''' and a metric is that a '''pseudometric''' does not insist that the distance function between distinct elements is ''strictly'' positive.
\end{definition}