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| In [[mathematics]], a '''pseudometric space''' 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]].
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| When a topology is generated using a family of pseudometrics, the space is called a [[gauge space]].
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| ==Definition==
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| 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 \mathbb{R}_{\geq 0}</math> (called a '''pseudometric''') such that, for every <math>x,y,z \in X</math>,
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| #<math>\,\!d(x,x) = 0</math>.
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| #<math>\,\!d(x,y) = d(y,x)</math> (''symmetry'')
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| #<math>\,\!d(x,z) \leq d(x,y) + d(y,z)</math> (''[[subadditivity]]''/''[[triangle inequality]]'')
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| <!-- Leave those \,\! at the start of each condition; see WP:MATH for why. -->
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| Unlike a metric space, points in a pseudometric space need not be [[identity of indiscernibles|distinguishable]]; that is, one may have <math>d(x,y)=0</math> for distinct values <math>x\ne y</math>.
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| ==Examples==
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| Pseudometrics arise naturally in [[functional analysis]]. Consider the space <math>\mathcal{F}(X)</math> of real-valued functions <math>f:X\to\mathbb{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
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| :<math>\,\!d(f,g) = |f(x_0)-g(x_0)|\;</math>
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| for <math>f,g\in \mathcal{F}(X)</math>
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| For vector spaces ''V'', a [[seminorm]] ''p'' induces a pseudometric on ''V'', as
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| :<math>\,\!d(x,y)=p(x-y).</math>
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| Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.
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| Pseudometrics also arise in the theory of hyperbolic [[complex manifold]]s: see [[Kobayashi metric]].
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| ==Topology==
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| The '''pseudometric topology''' is the [[topological space|topology]] induced by the [[open balls]]
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| :<math>B_r(p)=\{ x\in X\mid d(p,x)<r \},</math> | |
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| which form a [[basis (topology)|basis]] for the topology.<ref>{{planetmath reference|id=6284|title=Pseudometric topology}}</ref> 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.
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| 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 space|T<sub>0</sub>]] (i.e. distinct points are topologically distinguishable).
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| ==Metric identification==
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| 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> and let
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| :<math>d^*([x],[y])=d(x,y)</math> | |
| 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: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>
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| 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>.
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| An example of this construction is the [[Complete_metric_space#Completion|completion of a metric space]] by its [[Cauchy sequences]].
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite book | title=General Topology I: Basic Concepts and Constructions Dimension Theory | last=Arkhangel'skii | first=A.V. | coauthors=Pontryagin, L.S. | year=1990 | isbn=3-540-18178-4 | publisher=[[Springer Science+Business Media|Springer]] | series=Encyclopaedia of Mathematical Sciences}}
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| * {{cite book | title=Counterexamples in Topology | last=Steen | first=Lynn Arthur | coauthors=Seebach, Arthur | year=1995 | origyear=1970 | isbn=0-486-68735-X | publisher=[[Dover Publications]] | edition=new edition }}
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| * {{PlanetMath attribution|id=6273|title=Pseudometric space}}
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| * {{planetmath reference|id=6275|title=Example of pseudometric space}}
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| {{DEFAULTSORT:Pseudometric Space}}
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| [[Category:Properties of topological spaces]]
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| [[Category:Metric geometry]]
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Not much to tell about myself really.
Enjoying to be a part of wmflabs.org.
I just wish Im useful in one way here.
Here is my blog post :: lose weight fast