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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