# 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:

\((M1)\) | $:$ | \(\displaystyle \forall x \in A:\) | \(\displaystyle d \left({x, x}\right) = 0 \) | |||||

\((M2)\) | $:$ | \(\displaystyle \forall x, y, z \in A:\) | \(\displaystyle d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right) \) | |||||

\((M4)\) | $:$ | \(\displaystyle \forall x, y \in A:\) | \(\displaystyle x \ne y \implies d \left({x, y}\right) > 0 \) |

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

### Quasimetric Space

A **quasimetric space** $M = \left({A, d}\right)$ is an ordered pair consisting of a set $A \ne \varnothing$ 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

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 5$