# Pseudometric space

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

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

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

## Examples

• Pseudometrics arise naturally in functional analysis. Consider the space $\mathcal{F}(X)$ of real-valued functions $f\colon X\to\mathbb{R}$ together with a special point $x_0\in X$. This point then induces a pseudometric on the space of functions, given by
$d(f,g) = |f(x_0)-g(x_0)|$
for $f,g\in \mathcal{F}(X)$
• For vector spaces $V$, a seminorm $p$ induces a pseudometric on $V$, as
$d(x,y)=p(x-y).$
Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.
• Every measure space $(\Omega,\mathcal{A},\mu)$ can be viewed as a complete pseudometric space by defining
$d(A,B) := \mu(A\Delta B)$
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.[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 $x\sim y$ if $d(x,y)=0$. Let $X^*=X/{\sim}$ and let

$d^*([x],[y])=d(x,y)$

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

The metric identification preserves the induced topologies. That is, a subset $A\subset X$ is open (or closed) in $(X,d)$ if and only if $\pi(A)=[A]$ is open (or closed) in $(X^*,d^*)$. 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 $(X,d)$ be a pseudo-metric space and define an equivalence relation $\sim$ in $X$ by $x \sim y$ if $d(x,y)=0$. Let $Y$ be the quotient space $X/\sim$ and $p\colon X\to Y$ the canonical projection that maps each point of $X$ onto the equivalence class that contains it. Define the metric $\rho$ in $Y$ by $\rho(a,b) = d(p^{-1}(a),p^{-1}(b))$ for each pair $a,b \in Y$. It is easily shown that $\rho$ is indeed a metric and $\rho$ defines the quotient topology on $Y$."