# 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**.

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces