# Definition:Pseudometric

## Definition

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

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

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

\((M3)\) | $:$ | \(\displaystyle \forall x, y \in A:\) | \(\displaystyle d \left({x, y}\right) = d \left({y, x}\right) \) |

The difference between a **pseudometric** and a metric is that a **pseudometric** does not insist that the distance function between distinct elements is *strictly* positive.

### Pseudometric Space

A **pseudometric space** $M = \left({A, d}\right)$ is an ordered pair consisting of a set $A \ne \varnothing$ followed by a pseudometric $d: A \times A \to \R$ which acts on that set.

## Also known as

A **pseudometric** on a pseudometric space can be referred to as a distance function in the same way as a metric on a metric space.

## Also see

- Distance in Pseudometric is Non-Negative, where it is shown that $\forall x, y \in A: d \left({x, y}\right) \ge 0$ which is often taken as one of the axioms.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
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