# Definition:Pseudometric

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## Definition

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

\((\text M 1)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds \map d {x, x} = 0 \) | |||||

\((\text M 2)\) | $:$ | \(\ds \forall x, y, z \in A:\) | \(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \) | |||||

\((\text M 3)\) | $:$ | \(\ds \forall x, y \in A:\) | \(\ds \map d {x, y} = \map d {y, x} \) |

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** $M = \struct {A, d}$ is an ordered pair consisting of a set $A \ne \O$ 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 $\forall x, y \in A: \map d {x, y} \ge 0$ which is often taken as one of the axioms.

## 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