Pseudometric space

In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis.
When a topology is generated using a family of pseudometrics, the space is called a gauge space.
Definition.
A pseudometric space formula_1 is a set formula_2 together with a non-negative real-valued function formula_3 (called a pseudometric) such that,for every formula_4,
Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have formula_8 for distinct values formula_9.
Topology.
The pseudometric topology is the topology generated by the open balls
which form a basis for the topology. A topological space is said to be a pseudometrizable space if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space.
The difference between pseudometrics and metrics is entirely topological. That is, a pseudometric is a metric if and only if the topology it generates is T0 (i.e. distinct points are topologically distinguishable).
The definitions of Cauchy sequences and metric completion for metric spaces carry over to pseudometric spaces unchanged.
Metric identification.
The vanishing of the pseudometric induces an equivalence relation, called the metric identification, that converts the pseudometric space into a full-fledged metric space. This is done by defining formula_25 if formula_8. Let formula_27 be the quotient space of by this equivalence relation and define
Then formula_29 is a metric on formula_30 and formula_31 is a well-defined metric space, called the metric space induced by the pseudometric space formula_32.
The metric identification preserves the induced topologies. That is, a subset formula_33 is open (or closed) in formula_1 if and only if formula_35 is open (or closed) in formula_31 and "A" is saturated. The topological identification is the Kolmogorov quotient.
An example of this construction is the completion of a metric space by its Cauchy sequences.