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In [[mathematics]], a '''metric space aimed at its subspace''' is a [[category theory|categorical]] construction that has a direct geometric meaning.  It is also a useful step toward the construction of the ''metric envelope'', or [[tight span]], which are basic (injective) objects of the category of [[metric space]]s.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


Following {{harv|Holsztyński|1966}}, a notion of a metric space ''Y'' aimed at its subspace ''X'' is defined.
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]]
* 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.


Informally, imagine terrain Y, and its part X, such that wherever in Y you place a sharpshooter, and an apple at another place in Y, and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of X, or at least it will fly arbitrarily close to points of X – then we say that Y is aimed at X.
Registered users will be able to choose between the following three rendering modes:


A priori, it may seem plausible that for a given ''X'' the superspaces ''Y'' that aim at ''X'' can be arbitrarily large or at least huge.  We will see that this is not the case. Among the spaces, which aim at a subspace isometric to ''X'' there is a unique ([[up to]] [[isometry]]) universal one, Aim(''X''), which in a sense of canonical [[isometric embedding]]s contains any other space aimed at (an isometric image of) ''X''. And in the special case of an arbitrary compact metric space ''X'' every bounded subspace of an arbitrary metric space ''Y'' aimed at ''X'' is [[totally bounded]] (i.e. its metric completion is compact).
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


== Definitions ==
<!--'''PNG'''  (currently default in production)
Let <math>(Y, d)</math> be a metric space. Let <math>X</math> be a subset of <math>Y</math>, so that <math>(X,d |_X)</math> (the set <math>X</math> with the metric from <math>Y</math> restricted to <math>X</math>) is a metric subspace of <math>(Y,d)</math>.  Then
:<math forcemathmode="png">E=mc^2</math>


'''Definition'''.&nbsp; Space <math>Y</math> aims at <math>X</math> if and only if, for all points <math>y, z</math> of <math>Y</math>, and for every real <math>\epsilon > 0</math>, there exists a point <math>p</math> of <math>X</math> such that
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


:<math>|d(p,y) - d(p,z)| > d(y,z) - \epsilon.</math>
<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].


Let <math>\text{Met}(X)</math> be the space of all real valued [[metric map]]s (non-contractive) of <math>X</math>. Define
==Demos==


:<math>\text{Aim}(X) := \{f \in \operatorname{Met}(X) : f(p) + f(q) \ge d(p,q) \text{ for all } p,q\in X\}.</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


Then


:<math>d(f,g) := \sup_{x\in X} |f(x)-g(x)| < \infty</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.


for every <math>f, g\in \text{Aim}(X)</math> is a metric on <math>\text{Aim}(X)</math>. Furthermore, <math>\delta_X\colon x\mapsto d_x</math>, where <math>d_x(p) := d(x,p)\,</math>, is an isometric embedding of <math>X</math> into <math>\operatorname{Aim}(X)</math>; this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces <math>X</math> into <math>C(X)</math>, where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space <math>\operatorname{Aim}(X)</math> is aimed at <math>\delta_X(X)</math>.
==Test pages ==


== Properties ==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
Let <math>i\colon X \to Y</math> be an isometric embedding. Then there exists a natural metric map <math>j\colon Y \to \operatorname{Aim}(X)</math> such that <math>j \circ i = \delta_X</math>:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


:::<math>(j(y))(x) := d(x,y)\,</math>
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
for every <math>x\in X\,</math> and <math>y\in Y\,</math>.
==Bug reporting==
 
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 .
:'''Theorem''' The space ''Y'' above is aimed at subspace ''X'' if and only if the natural mapping <math>j\colon Y \to \operatorname{Aim}(X)</math> is an isometric embedding.
 
Thus it follows that every space aimed at ''X'' can be isometrically mapped into Aim(X), with some additional (essential) categorical requirements satisfied.
 
The space Aim(X) is [[injective metric space|injective]] (hyperconvex in the sense of [[Aronszajn]]-Panitchpakdi) – given a metric space ''M,'' which contains Aim(X) as a metric subspace, there is a canonical (and explicit) metric retraction of ''M'' onto Aim(X) {{harv|Holsztyński|1966}}.
 
==References==
*{{citation|mr=0196709|last= Holsztyński|first= W.|title= On metric spaces aimed at their subspaces. |journal= Prace Mat.|volume= 10|year= 1966|pages= 95–100}}
 
[[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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