View source for Pseudometric space

{{Short description|Generalization of metric spaces in mathematics}}
In [[Mathematics|mathematics]], a '''pseudometric space''' is a [[Generalization|generalization]] of a [[Metric space|metric space]] in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by [[Biography:Đuro Kurepa|Đuro Kurepa]]<ref>{{Cite journal|last=Kurepa|first=Đuro|date=1934|title=Tableaux ramifiés d'ensembles, espaces pseudodistaciés|journal=C. R. Acad. Sci. Paris|volume=198 (1934)|pages=1563–1565}}</ref><ref>{{Cite book|last=Collatz|first=Lothar|title=Functional Analysis and Numerical Mathematics|publisher=[[Company:Academic Press|Academic Press]]|year=1966|location=New York, San Francisco, London|pages=51|language=English}}</ref> in 1934. In the same way as every normed space is a [[Metric space|metric space]], every seminormed space is a pseudometric space. Because of this analogy, the term semimetric space (which has a different meaning in [[Topology|topology]]) is sometimes used as a synonym, especially in [[Functional analysis|functional analysis]].

When a topology is generated using a family of pseudometrics, the space is called a gauge space.

==Definition==

A pseudometric space <math>(X,d)</math> is a set <math>X</math> together with a non-negative [[Real-valued function|real-valued function]] <math>d : X \times X \longrightarrow \R_{\geq 0},</math> called a '''{{visible anchor|pseudometric}}''', such that for every <math>x, y, z \in X,</math>

#<math>d(x,x) = 0.</math>
#''Symmetry'': <math>d(x,y) = d(y,x)</math>
#''[[Subadditivity]]''/''[[Triangle inequality]]'': <math>d(x,z) \leq d(x,y) + d(y,z)</math>
Unlike a metric space, points in a pseudometric space need not be [[Philosophy:Identity of indiscernibles|distinguishable]]; that is, one may have <math>d(x, y) = 0</math> for distinct values <math>x \neq y.</math>

==Examples==

Any metric space is a pseudometric space. 
Pseudometrics arise naturally in [[Functional analysis|functional analysis]]. Consider the space <math>\mathcal{F}(X)</math> of real-valued functions <math>f : X \to \R</math> together with a special point <math>x_0 \in X.</math> This point then induces a pseudometric on the space of functions, given by <math display=block>d(f,g) = \left|f(x_0) - g(x_0)\right|</math> for <math>f, g \in \mathcal{F}(X)</math>

A [[Seminorm|seminorm]] <math>p</math> induces the pseudometric <math>d(x, y) = p(x - y)</math>. This is a [[Convex function|convex function]] of an affine function of <math>x</math> (in particular, a [[Translation (geometry)|translation]]), and therefore convex in <math>x</math>. (Likewise for <math>y</math>.)

Conversely, a homogeneous, translation-invariant pseudometric induces a seminorm.

Pseudometrics also arise in the theory of [[Hyperbolic manifold|hyperbolic]] [[Complex manifold|complex manifold]]s: see [[Kobayashi metric]]. 

Every [[Measure space|measure space]] <math>(\Omega,\mathcal{A},\mu)</math> can be viewed as a complete pseudometric space by defining <math display=block>d(A,B) := \mu(A \vartriangle B)</math> for all <math>A, B \in \mathcal{A},</math> where the triangle denotes [[Symmetric difference|symmetric difference]].

If <math>f : X_1 \to X_2</math> is a function and ''d''<sub>2</sub> is a pseudometric on ''X''<sub>2</sub>, then <math>d_1(x, y) := d_2(f(x), f(y))</math> gives a pseudometric on ''X''<sub>1</sub>. If ''d''<sub>2</sub> is a metric and ''f'' is [[Injective function|injective]], then ''d''<sub>1</sub> is a metric.

==Topology==

The '''{{visible anchor|pseudometric topology}}''' is the topology generated by the open balls
<math display=block>B_r(p) = \{x \in X : d(p, x) < r\},</math>
which form a basis for the topology.<ref>{{planetmath reference|urlname=PseudometricTopology|title=Pseudometric topology}}</ref> A topological space is said to be a '''{{visible anchor|pseudometrizable space}}'''<ref>Willard, p. 23</ref> 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 T<sub>0</sub> (that is, distinct points are topologically distinguishable).

The definitions of [[Cauchy sequence]]s and metric completion for metric spaces carry over to pseudometric spaces unchanged.<ref>{{Cite web|last=Cain|first=George|date=Summer 2000|title=Chapter 7: Complete pseudometric spaces|url=http://people.math.gatech.edu/~cain/summer00/ch7.pdf|url-status=live|archive-url=https://archive.today/fnt7f|archive-date=7 October 2020|access-date=7 October 2020}}</ref>

==Metric identification==

The vanishing of the pseudometric induces an [[Equivalence relation|equivalence relation]], called the '''metric identification''', that converts the pseudometric space into a full-fledged [[Metric space|metric space]].  This is done by defining <math>x\sim y</math> if <math>d(x,y)=0</math>. Let <math>X^* = X/{\sim}</math> be the [[Quotient space (topology)|quotient space]] of <math>X</math> by this equivalence relation and define
<math display=block>\begin{align}
  d^*:(X/\sim)&\times (X/\sim) \longrightarrow \R_{\geq 0} \\
  d^*([x],[y])&=d(x,y)
\end{align}</math>
This is well defined because for any <math>x' \in [x]</math> we have that <math>d(x, x') = 0</math> and so <math>d(x', y) \leq d(x, x') + d(x, y) = d(x, y)</math> and vice versa. Then <math>d^*</math> is a metric on <math>X^*</math> and <math>(X^*,d^*)</math> is a well-defined metric space, called the '''metric space induced by the pseudometric space''' <math>(X, d)</math>.<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=https://www.springer.com/mathematics/analysis/book/978-0-387-97986-1|access-date=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 : 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><ref>{{cite book|title=A comprehensive course in analysis|last=Simon|first=Barry|publisher=American Mathematical Society|year=2015|isbn=978-1470410995|location=Providence, Rhode Island}}</ref>

The metric identification preserves the induced topologies. That is, a subset <math>A \subseteq 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>\left(X^*, d^*\right)</math> and <math>A</math> is [[Saturated set|saturated]]. 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.

==See also==

* {{annotated link|Generalised metric}}
* {{annotated link|Metric signature}}
* {{annotated link|Metric space}}
* {{annotated link|Metrizable topological vector space}}

==Notes==

{{reflist}}

==References==

* {{cite book | title=General Topology I: Basic Concepts and Constructions Dimension Theory | last=Arkhangel'skii | first=A.V. |author1link = Alexander Arhangelskii|author2=Pontryagin, L.S.  |author2link = Lev Pontryagin| year=1990 | isbn=3-540-18178-4 | publisher=[[Company:Springer Science+Business Media|Springer]] | series=Encyclopaedia of Mathematical Sciences}}
* {{cite book | title=Counterexamples in Topology | last=Steen | first=Lynn Arthur |author1link = Lynn Arthur Steen|author2link = J. Arthur Seebach Jr.|author2=Seebach, Arthur  | year=1995 | orig-year=1970 | isbn=0-486-68735-X | publisher=[[Social:Dover Publications|Dover Publications]] | edition=new }}
* {{Citation | last=Willard | first=Stephen | title=General Topology | orig-year=1970 | publisher=Addison-Wesley | edition=[[Social:Dover Publications|Dover]] reprint of 1970 | year=2004}}
* {{planetmath reference|urlname=ExampleOfPseudometricSpace|title=Example of pseudometric space}}

{{DEFAULTSORT:Pseudometric Space}}

[[Category:Properties of topological spaces]]
[[Category:Metric geometry]]

{{Sourceattribution|Pseudometric space}}