# Pseudometric space

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In mathematics, a pseudometric or semi-metric space[1] 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

A pseudometric space ${\displaystyle (X,d)}$ is a set ${\displaystyle X}$ together with a non-negative real-valued function ${\displaystyle d\colon X\times X\longrightarrow \mathbb {R} _{\geq 0}}$ (called a pseudometric) such that, for every ${\displaystyle x,y,z\in X}$,

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

## Examples

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

## Topology

The pseudometric topology is the topology induced by the open balls

${\displaystyle B_{r}(p)=\{x\in X\mid d(p,x)

which form a basis for the topology.[2] 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 ${\displaystyle x\sim y}$ if ${\displaystyle d(x,y)=0}$. Let ${\displaystyle X^{*}=X/{\sim }}$ and let

${\displaystyle d^{*}([x],[y])=d(x,y)}$

Then ${\displaystyle d^{*}}$ is a metric on ${\displaystyle X^{*}}$ and ${\displaystyle (X^{*},d^{*})}$ is a well-defined metric space.[3]

The metric identification preserves the induced topologies. That is, a subset ${\displaystyle A\subset X}$ is open (or closed) in ${\displaystyle (X,d)}$ if and only if ${\displaystyle \pi (A)=[A]}$ is open (or closed) in ${\displaystyle (X^{*},d^{*})}$. The topological identification is the Kolmogorov quotient.

An example of this construction is the completion of a metric space by its Cauchy sequences.

## Notes

1. Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0-8218-2129-6.
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## References

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