Definition:Pseudometric

Definition
A pseudo-metric on a set $$X$$ is a real-valued function (called the distance function or simply distance) $$d: X \times X \to \R$$ which satisfies:


 * $$\forall x, y \in X: d \left({x, y}\right) \ge 0$$
 * $$\forall x, y \in X: x = y \implies d \left({x, y}\right) = 0$$
 * $$\forall x, y \in X: d \left({x, y}\right) = d \left({y, x}\right)$$
 * $$\forall x, y, z \in X: d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right)$$

Also see
Compare this definition with that for a metric.

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