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: $d \left({x, y}\right) = d \left({y, x}\right)$

Distance Function
The function $d: X \times X \to \R$ is called the distance function or simply distance.

Pseudometric Space
A pseudometric space $M = \left({A, d}\right)$ is an ordered pair consisting of a set $A \ne \varnothing$ followed by a pseudometric $d: A \times A \to \R$ which acts on that set.

The elements of $A$ are called the points of the space.

Comparison with Metric Space

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


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

which is often taken as one of the axioms.


 * 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.

Also see

 * Quasimetric