# 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 (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. $d(x,z)\leqslant d(x,y)+d(y,z)$  (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 $x\neq y$ .

## 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.
• Pseudometrics also arise in the theory of hyperbolic complex manifolds: see Kobayashi metric.
• Every measure space $(\Omega ,{\mathcal {A}},\mu )$  can be viewed as a complete pseudometric space by defining
$d(A,B):=\mu (A\vartriangle B)$
for all $A,B\in {\mathcal {A}}$ , where the triangle denotes symmetric difference.
• If $f:X_{1}\rightarrow X_{2}$  is a function and d2 is a pseudometric on X2, then $d_{1}(x,y):=d_{2}(f(x),f(y))$  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

$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 $X^{*}=X/{\sim }$  be the quotient space of X by this equivalence relation and define

{\begin{aligned}d^{*}:(X/\sim )&\times (X/\sim )\longrightarrow \mathbb {R} _{\geq 0}\\d^{*}([x],[y])&=d(x,y)\end{aligned}}

Then $d^{*}$  is a metric on $X^{*}$  and $(X^{*},d^{*})$  is a well-defined metric space, called the metric space induced by the pseudometric space $(X,d)$ .

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