Definition:Quasimetric

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

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