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 $x,y,z\in X$ ,

$d(x,x)=0$ .
$d(x,y)=d(y,x)$ (symmetry)
$d(x,z)\leqslant d(x,y)+d(y,z)$ (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)<r\},

which form a basis for the topology.^[1] 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 $(X,d)$ .^[2]^[3]

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.

Notes

^ "Pseudometric topology". PlanetMath.
^ 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 $Y$ .
^ Simon, Barry (2015). A comprehensive course in analysis. Providence, Rhode Island: American Mathematical Society. ISBN 978-1470410995.

References

Arkhangel'skii, A.V.; Pontryagin, L.S. (1990). General Topology I: Basic Concepts and Constructions Dimension Theory. Encyclopaedia of Mathematical Sciences. Springer. ISBN 3-540-18178-4.
Steen, Lynn Arthur; Seebach, Arthur (1995) [1970]. Counterexamples in Topology (new ed.). Dover Publications. ISBN 0-486-68735-X.
This article incorporates material from Pseudometric space on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
"Example of pseudometric space". PlanetMath.

[1] "Pseudometric topology". PlanetMath.

[2] 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 $Y$ .

[3] Simon, Barry (2015). A comprehensive course in analysis. Providence, Rhode Island: American Mathematical Society. ISBN 978-1470410995.

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