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In [[mathematics]], '''pointless topology''' (also called '''point-free''' or '''pointfree topology''') is an approach to [[topology]] that avoids mentioning points. The name 'pointless topology' is due to [[John von Neumann]].<ref>Garrett Birkhoff, ''VON NEUMANN AND LATTICE THEORY'', ''John Von Neumann 1903-1957'', J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5 </ref>  The ideas of pointless topology are closely related to [[mereotopology| mereotopologies]] in which regions (sets) are treated as foundational without explicit reference to underlying point sets.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


==General concepts==
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]]
Traditionally, a [[topological space]] consists of a [[Set (mathematics)|set]] of [[point (topology)|points]], together with a system of [[open set]]s. These open sets with the operations of [[intersection (set theory)|intersection]] and [[union (set theory)|union]] form a [[lattice (order)|lattice]] with certain properties. Pointless topology then studies lattices like these abstractly, without reference to any underlying set of points. Since some of the so-defined lattices do not arise from topological spaces, one may see the [[category theory|category]] of pointless topological spaces, also called [[Frames and locales|locales]], as an extension of the category of ordinary topological spaces.
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==Categories of frames and locales==
Registered users will be able to choose between the following three rendering modes:
Formally, a '''frame''' is defined to be a [[lattice (order)|lattice]] ''L'' in which  finite [[meet]]s [[Distributivity (order theory)|distribute]] over arbitrary [[join]]s, i.e. every (even infinite) subset {''a''<sub>i</sub>} of ''L'' has a [[supremum]] ⋁''a''<sub>''i''</sub> such that


:<math>b \wedge \left( \bigvee a_i\right) = \bigvee \left(a_i \wedge b\right)</math>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


for all ''b'' in ''L''. These frames, together with lattice homomorphisms that respect arbitrary suprema, form a category. The [[dual (category theory)|dual]] of the '''category of frames''' is called the '''category of locales''' and generalizes the category '''[[category of topological spaces|Top]]''' of all topological spaces with continuous functions. The consideration of the dual category is motivated by the fact that every [[continuous function (topology)|continuous map]] between topological spaces ''X'' and ''Y'' induces a map between the lattices of open sets ''in the opposite direction'' as for every continuous function ''f'':&nbsp;''X''&nbsp;&rarr;&nbsp;''Y'' and every open set ''O'' in ''Y'' the [[inverse image]] ''f''<sup>&nbsp;-1</sup>(''O'') is an open set in ''X''.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


==Relation to point-set topology==
'''source'''
It is possible to translate most concepts of point-set topology into the context of locales, and prove analogous theorems. While many important theorems in point-set topology require the [[axiom of choice]], this is not true for some of their analogues in locale theory. This can be useful if one works in a [[topos]] that does not have the axiom of choice.
:<math forcemathmode="source">E=mc^2</math> -->


The concept of "product of locales" diverges slightly from the concept of "[[Product_topology|product of topological spaces]]", and this divergence has been called a disadvantage of the locale approach.
<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].
Others{{who|date=November 2010}} claim that the locale product is more natural, and point to several "desirable" properties{{Which?|date=November 2010}} not shared by products of topological spaces.


For almost all spaces (more precisely for [[sober space]]s), the topological product and the localic product have the same set of points. The products differ in how equality between sets of open rectangles, the canonical base for the product topology, is defined: equality for the topological product means the same set of points is covered;
==Demos==
equality for the localic product means provable equality using the frame axioms. As a result, two open sublocales of a localic product may contain exactly the same points without being equal.


A point where locale theory and topology diverge much more strongly is the concept of subspaces vs. sublocales.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
The rational numbers have ''c'' subspaces but 2<sup>''c''</sup> sublocales. The proof for the latter statement is due to [[John Isbell]], and uses the fact that the rational numbers have ''c'' many pairwise almost disjoint (= finite intersection) closed subspaces.


==See also==
* [[Heyting algebra]]. A locale is a [[complete Heyting algebra]].
* Details on the relationship between the category of topological spaces and the category of locales, including the explicit construction of the duality between [[sober space]]s and spatial locales, are to be found in the article on [[Stone duality]].
* [[Point-free geometry]]
* [[Mereology]]
* [[Mereotopology]]
* [[Tacit programming]]


==References==
* accessibility:
{{reflist}}
** 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]]
*[[Peter Johnstone (mathematician)|Johnstone, Peter T.]], 1983, "[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183550014 The point of pointless topology,]" ''Bulletin of the American Mathematical Society 8(1)'': 41-53.
** [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.


[[Category:Category theory]]
==Test pages ==
[[Category:General topology]]


[[es:Topología sin puntos]]
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
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*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==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 .

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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