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In [[mathematics]], a '''pseudometric''' or  '''semi-metric space'''<ref>Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0-8218-2129-6.</ref> 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]].
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


When a topology is generated using a family of pseudometrics, the space is called a [[gauge space]].
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
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==Definition==
Registered users will be able to choose between the following three rendering modes:
A pseudometric space <math>(X,d)</math> is a set <math>X</math> together with a non-negative real-valued function <math>d\colon X \times X \longrightarrow \mathbb{R}_{\geq 0}</math> (called a '''pseudometric''') such that, for every <math>x,y,z \in X</math>,


#<math>d(x,x) = 0</math>.
'''MathML'''
#<math>d(x,y) = d(y,x)</math> (''symmetry'')
:<math forcemathmode="mathml">E=mc^2</math>
#<math>d(x,z) \leq d(x,y) + d(y,z)</math> (''[[subadditivity]]''/''[[triangle inequality]]'')
<!-- \,\!  not useful anymore, hence not mentioned in WP:MATH. -->


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>.
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


==Examples==
'''source'''
* Pseudometrics arise naturally in [[functional analysis]]. Consider the space <math>\mathcal{F}(X)</math> of real-valued functions <math>f\colon 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
:<math forcemathmode="source">E=mc^2</math> -->
::<math>d(f,g) = |f(x_0)-g(x_0)|</math>
:for <math>f,g\in \mathcal{F}(X)</math>


* For vector spaces <math>V</math>, a [[seminorm]] <math>p</math> induces a pseudometric on <math>V</math>, as
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
::<math>d(x,y)=p(x-y).</math>
:Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.


* Pseudometrics also arise in the theory of hyperbolic [[complex manifold]]s: see [[Kobayashi metric]].
==Demos==


* Every [[measure space]] <math>(\Omega,\mathcal{A},\mu)</math> can be viewed as a complete pseudometric space by defining
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
::<math>d(A,B) := \mu(A\Delta B)</math>
:for all <math>A,B\in\mathcal{A}</math>.
* If <math>f:X_1 \rightarrow 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 '''pseudometric topology''' is the [[topological space|topology]] induced by the [[open balls]]


:<math>B_r(p)=\{ x\in X\mid d(p,x)<r \},</math>
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


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.
==Test pages ==


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).
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


==Metric identification==
*[[Inputtypes|Inputtypes (private Wikis only)]]
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
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>d^*([x],[y])=d(x,y)</math>
==Bug reporting==
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>
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
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]].
 
==Notes==
{{Reflist}}
 
==References==
* {{cite book | title=General Topology I: Basic Concepts and Constructions Dimension Theory | last=Arkhangel'skii | first=A.V. |author2=Pontryagin, L.S.  | year=1990 | isbn=3-540-18178-4 | publisher=[[Springer Science+Business Media|Springer]] | series=Encyclopaedia of Mathematical Sciences}}
* {{cite book | title=Counterexamples in Topology | last=Steen | first=Lynn Arthur |author2=Seebach, Arthur  | year=1995 | origyear=1970 | isbn=0-486-68735-X | publisher=[[Dover Publications]] | edition=new edition }}
* {{PlanetMath attribution|id=6273|title=Pseudometric space}}
* {{planetmath reference|id=6275|title=Example of pseudometric space}}
 
{{DEFAULTSORT:Pseudometric Space}}
[[Category:Properties of topological spaces]]
[[Category:Metric geometry]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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