Pseudometric space

{{Short description|Generalization of metric spaces in mathematics}} {{About|spaces with a nonnegative distance function|manifolds with a Pseudo-Riemannian metric tensor|Pseudo-Riemannian manifold}} In mathematics, a '''pseudometric space''' is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by 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=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, 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 <math>(X,d)</math> is a set <math>X</math> together with a non-negative 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 distinguishable; that is, one may have <math>d(x, y) = 0</math> for distinct values <math>x \neq y.</math>

It worth while noting that Symmetry and classical Triangle inequality can be replaced with single modified Triangle one: <math>d(x,y) \leq d(x,z) + d(y,z)</math>. This inequality combines symmetry and classical Triangle inequality.

==Examples==

Any metric space is a pseudometric space. Pseudometrics arise naturally in 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 <math>p</math> induces the pseudometric <math>d(x, y) = p(x - y)</math>. This is a convex function of an affine function of <math>x</math> (in particular, a 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 complex manifolds: see Kobayashi metric.

Every 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.

If <math>f : X_1 \to X_2</math> is a function and ''d''2 is a pseudometric on ''X''2, then <math>d_1(x, y) := d_2(f(x), f(y))</math> gives a pseudometric on ''X''1. If ''d''2 is a metric and ''f'' is injective, then ''d''1 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 T0 (that is, distinct points are topologically distinguishable).

The definitions of Cauchy sequences 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/20201007070509/http://people.math.gatech.edu/~cain/summer00/ch7.pdf|archive-date=7 October 2020|access-date=7 October 2020}}</ref>

==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 <math>x\sim y</math> if <math>d(x,y)=0</math>. Let <math>X^* = X/{\sim}</math> be the 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. The topological identification is the Kolmogorov quotient.

An example of this construction is the completion of a metric space by its Cauchy sequences.

== Properties ==

* In a pseudometric space, the set of all points (say, X) which are of distance 0 relative to another set Y in the space is the closure of Y, or cl(Y) = x.<ref>{{Cite book |last=Kelley |first=John L. |title=General Topology |date=2017 |publisher=Dover Publications |isbn=978-0-486-81544-2 |series=Dover Books on Mathematics |location=Mineola |pages=120}}</ref>

==See also==

* {{annotated link|Generalised metric}} * {{annotated link|Uniform space}} * {{annotated link|Pseudo-Riemannian manifold}} * {{annotated link|Metric space}} * {{annotated link|Metrizable topological vector space}}

==Notes==

==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=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=Dover Publications | edition=new }} * {{Citation | last=Willard | first=Stephen | title=General Topology | orig-year=1970 | publisher=Addison-Wesley | edition=Dover reprint of 1970 | year=2004}} * {{PlanetMath attribution|id=6273|title=Pseudometric space}} * {{planetmath reference|urlname=ExampleOfPseudometricSpace|title=Example of pseudometric space}}

Category:Metric geometry Category:Properties of topological spaces