Definition:Pseudometric

Definition
A pseudometric on a set $A$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions:

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.

Also see

 * Definition:Metric
 * Definition:Quasimetric


 * Distance in Pseudometric is Non-Negative, where it is shown that $\forall x, y \in A: \map d {x, y} \ge 0$ which is often taken as one of the axioms.