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 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 is a set together with a non-negative real-valued function such that,for every,
  1. .
Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have for distinct values.

Examples

The pseudometric topology is the topology induced by the open balls
which form a basis for the topology. A topological space is said to be a pseudometrizable topological 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.

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 if. Let be the quotient space of by this equivalence relation and define
Then is a metric on and is a well-defined metric space, called the metric space induced by the pseudometric space.
The metric identification preserves the induced topologies. That is, a subset is open in if and only if is open in 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.