Definition:Pseudometric

Definition
A pseudometric on a set $X$ is a real-valued function $d: X \times X \to \R$ which satisfies the following conditions for all $x, y, z \in X$:


 * M1: $d \left({x, x}\right) = 0$
 * M2: $d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right)$
 * M3: $d \left({x, y}\right) = d \left({y, x}\right)$

Also see
From Distance in Pseudometric Non-Negative, it can be seen that:

which is often taken as one of the axioms.
 * $\forall x, y \in X: d \left({x, y}\right) \ge 0$

Compare this definition with that for a metric.

The difference between a pseudometric and a metric is that a pseudometric does not insist that the distance function between distinct points is strictly positive.