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 points is strictly positive.

Also known as
A pseudometric on a pseudometric space can be referred to as a distance function in the same way as a metric on a metric space.

Also see

 * Metric
 * Quasimetric


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