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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:

\((\text M 1)\)   $:$     \(\ds \forall x \in A:\) \(\ds \map d {x, x} = 0 \)      
\((\text M 2)\)   $:$     \(\ds \forall x, y, z \in A:\) \(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \)      
\((\text M 3)\)   $:$     \(\ds \forall x, y \in A:\) \(\ds \map d {x, y} = \map d {y, x} \)      

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.

Some sources hyphenate: pseudo-metric.

Also see

  • Results about pseudometric spaces can be found here.