Pseudometric

A pseudometric on a set is a nonnegative function satisfying , symmetry , and the triangle inequality for all (Knapp 2016, p. 84). Unlike a metric, it allows the distance between two distinct points to be zero.

An example of a pseudometric on the set of all functions is defined by . It is nonnegative, symmetric, fulfills the triangle inequality and the condition , but it is also true that .

A pseudometric determines an equivalence relation by iff . On the set of equivalence classes, the formula

defines a metric. The triangle inequality ensures that this value does not depend on the representatives chosen for and . Thus identifying points at zero distance gives a metric space (Knapp 2016, p. 95).