# Pseudometric space

In mathematics, a pseudometric space 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 ${\displaystyle x,y,z\in X}$,

1. ${\displaystyle d(x,x)=0}$.
2. ${\displaystyle d(x,y)=d(y,x)}$ (symmetry)
3. ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$ (subadditivity/triangle inequality)

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

• Pseudometrics arise naturally in functional analysis. Consider the space ${\displaystyle {\mathcal {F}}(X)}$ of real-valued functions ${\displaystyle f\colon X\to \mathbb {R} }$ together with a special point ${\displaystyle x_{0}\in X}$. This point then induces a pseudometric on the space of functions, given by
${\displaystyle d(f,g)=|f(x_{0})-g(x_{0})|}$
for ${\displaystyle f,g\in {\mathcal {F}}(X)}$
• For vector spaces ${\displaystyle V}$, a seminorm ${\displaystyle p}$ induces a pseudometric on ${\displaystyle V}$, as
${\displaystyle d(x,y)=p(x-y).}$
Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.
• Pseudometrics also arise in the theory of hyperbolic complex manifolds: see Kobayashi metric.
• Every measure space ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ can be viewed as a complete pseudometric space by defining
${\displaystyle d(A,B):=\mu (A\vartriangle B)}$
for all ${\displaystyle A,B\in {\mathcal {A}}}$, where the triangle denotes symmetric difference.
• If ${\displaystyle f:X_{1}\rightarrow X_{2}}$ is a function and d2 is a pseudometric on X2, then ${\displaystyle d_{1}(x,y):=d_{2}(f(x),f(y))}$ gives a pseudometric on X1. If d2 is a metric and f is injective, then d1 is a metric.

## 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, called the metric space induced by the pseudometric space ${\displaystyle (X,d)}$.[3][4]

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.
2. ^
3. ^ Howes, Norman R. (1995). Modern Analysis and Topology. New York, NY: Springer. p. 27. ISBN 0-387-97986-7. Retrieved 10 September 2012. Let ${\displaystyle (X,d)}$ be a pseudo-metric space and define an equivalence relation ${\displaystyle \sim }$ in ${\displaystyle X}$ by ${\displaystyle x\sim y}$ if ${\displaystyle d(x,y)=0}$. Let ${\displaystyle Y}$ be the quotient space ${\displaystyle X/\sim }$ and ${\displaystyle p\colon X\to Y}$ the canonical projection that maps each point of ${\displaystyle X}$ onto the equivalence class that contains it. Define the metric ${\displaystyle \rho }$ in ${\displaystyle Y}$ by ${\displaystyle \rho (a,b)=d(p^{-1}(a),p^{-1}(b))}$ for each pair ${\displaystyle a,b\in Y}$. It is easily shown that ${\displaystyle \rho }$ is indeed a metric and ${\displaystyle \rho }$ defines the quotient topology on ${\displaystyle Y}$.
4. ^ Simon, Barry (2015). A comprehensive course in analysis. Providence, Rhode Island: American Mathematical Society. ISBN 1470410990.