Definition:Quasimetric

Definition
Let $A$ be a set.

Let $d: A \times A \to \R$ be a real-valued function.

$d$ is a quasimetric on $A$ $d$ satisfies the quasimetric axioms:

Note the numbering system of these conditions. They are numbered this way so as to retain consistency with the metric space axioms, of which these are a subset.

The difference between a quasimetric and a metric is that a quasimetric does not insist that the distance function between distinct elements is commutative, that is, that $\map d {x, y} = \map d {y, x}$.

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

Also see

 * Definition:Metric
 * Definition:Pseudometric