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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 $(X,d)$ 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)
$d(x,z)\leq d(x,y)+d(y,z)$ (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 $x\neq y$ .

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\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)<r\},

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

^ Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0-8218-2129-6.
^ "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 $Y$ .

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 edition ed.). Dover Publications. ISBN 0-486-68735-X. {{cite book}}: |edition= has extra text (help)
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] Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0-8218-2129-6.

[2] "Pseudometric topology". PlanetMath.

[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 $Y$ .

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@@ Line 42: / Line 42: @@
 Then <math>d^*</math> is a metric on <math>X^*</math> and <math>(X^*,d^*)</math> is a well-defined metric space.<ref>{{cite book|last=Howes|first=Norman R.|title=Modern Analysis and Topology|year=1995|publisher=Springer|location=New York, NY|isbn=0-387-97986-7|url=http://www.springer.com/mathematics/analysis/book/978-0-387-97986-1|accessdate=10 September 2012|page=27|quote=Let <math>(X,d)</math> be a pseudo-metric space and define an equivalence relation <math>\sim</math> in <math>X</math> by <math>x \sim y</math> if <math>d(x,y)=0</math>. Let <math>Y</math> be the quotient space <math>X/\sim</math> and <math>p\colon X\to Y</math> the canonical projection that maps each point of <math>X</math> onto the equivalence class that contains it. Define the metric <math>\rho</math> in <math>Y</math> by <math>\rho(a,b) = d(p^{-1}(a),p^{-1}(b))</math> for each pair <math>a,b \in Y</math>. It is easily shown that <math>\rho</math> is indeed a metric and <math>\rho</math> defines the quotient topology on <math>Y</math>.}}</ref>
-The metric identification preserves the induced topologies. That is, a subset <math>A\subset X</math> is open (or closed) in <math>(X,d)</math> if and only if <math>\pi(A)=[A]</math> is open (or closed) in <math>(X^*,d^*)</math>.
+The metric identification preserves the induced topologies. That is, a subset <math>A\subset X</math> is open (or closed) in <math>(X,d)</math> if and only if <math>\pi(A)=[A]</math> is open (or closed) in <math>(X^*,d^*)</math>. The topological identification is the [[Kolmogorov quotient]].
 An example of this construction is the [[Complete_metric_space#Completion|completion of a metric space]] by its [[Cauchy sequences]].