# 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.