# Pseudometric space

In mathematics, a pseudometric or semi-metric space is a generalized 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

Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have $d(x,y)=0$ for distinct values $x\neq y$ .

## Examples

$d(f,g)=|f(x_{0})-g(x_{0})|$ for $f,g\in {\mathcal {F}}(X)$ $d(x,y)=p(x-y).$ Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.
$d(A,B):=\mu (A\Delta B)$ for all $A,B\in {\mathcal {A}}$ .

## Topology

The pseudometric topology is the topology induced by the open balls

$B_{r}(p)=\{x\in X\mid d(p,x) 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 (i.e. distinct points are topologically distinguishable).

## 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 $x\sim y$ if $d(x,y)=0$ . Let $X^{*}=X/{\sim }$ and let

$d^{*}([x],[y])=d(x,y)$ An example of this construction is the completion of a metric space by its Cauchy sequences.