# 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 $A$ if and only if $d$ satisfies the quasimetric axioms:

 $(\text M 1)$ $:$ $\ds \forall x \in A:$ $\ds \map d {x, x} = 0$ $(\text M 2)$ $:$ $\ds \forall x, y, z \in A:$ $\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z}$ $(\text M 4)$ $:$ $\ds \forall x, y \in A:$ $\ds x \ne y \implies \map d {x, y} > 0$

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 $\map d {x, y} = \map d {y, x}$.

### Quasimetric Space

A quasimetric space $M = \struct {A, d}$ is an ordered pair consisting of a set $A \ne \O$ followed by a quasimetric $d: A \times A \to \R$ which acts on that set.

## 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

• Results about quasimetric spaces can be found here.