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In [[mathematics]], the '''category of topological spaces''', often denoted '''Top''', is the [[category (category theory)|category]] whose [[object (category theory)|object]]s are [[topological space]]s and whose [[morphism]]s are [[continuous map]]s or some other variant; for example, objects are often assumed to be [[compactly generated space|compactly generated]]. This is a category because the [[function composition|composition]] of two continuous maps is again continuous. The study of '''Top''' and of properties of [[topological space]]s using the techniques of [[category theory]] is known as '''categorical topology'''.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


N.B. Some authors use the name '''Top''' for the category with [[topological manifold]]s as objects and continuous maps as morphisms.
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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* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


==As a concrete category==
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Like many categories, the category '''Top''' is a [[concrete category]] (also known as a ''construct''), meaning its objects are [[Set (mathematics)|sets]] with additional structure (i.e. topologies) and its morphisms are [[function (mathematics)|function]]s preserving this structure. There is a natural [[forgetful functor]]
'''MathML'''
:''U'' : '''Top''' → '''Set'''
:<math forcemathmode="mathml">E=mc^2</math>
to the [[category of sets]] which assigns to each topological space the underlying set and to each continuous map the underlying [[function (mathematics)|function]].


The forgetful functor ''U'' has both a [[left adjoint]]
<!--'''PNG''' (currently default in production)
:''D'' : '''Set''' &rarr; '''Top'''
:<math forcemathmode="png">E=mc^2</math>
which equips a given set with the [[discrete topology]] and a [[right adjoint]]
:''I'' : '''Set''' &rarr; '''Top'''
which equips a given set with the [[indiscrete topology]]. Both of these functors are, in fact, [[right inverse]]s to ''U'' (meaning that ''UD'' and ''UI'' are equal to the [[identity functor]] on '''Set'''). Moreover, since any function between discrete or indiscrete spaces is continuous, both of these functors give [[full embedding]]s of '''Set''' into '''Top'''.


The construct '''Top''' is also ''fiber-complete'' meaning that the [[lattice of topologies|category of all topologies]] on a given set ''X'' (called the ''[[fiber (mathematics)|fiber]]'' of ''U'' above ''X'') forms a [[complete lattice]] when ordered by [[set inclusion|inclusion]]. The [[greatest element]] in this fiber is the discrete topology on ''X'' while the [[least element]] is the indiscrete topology.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


The construct '''Top''' is the model of what is called a [[topological category]]. These categories are characterized by the fact that every [[structured source]] <math>(X \to UA_i)_I</math> has a unique [[initial lift]] <math>( A \to A_i)_I</math>. In '''Top''' the initial lift is obtained by placing the [[initial topology]] on the source. Topological categories have many properties in common with '''Top''' (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).
<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].


==Limits and colimits==
==Demos==


The category '''Top''' is both [[complete category|complete and cocomplete]], which means that all small [[limit (category theory)|limits and colimit]]s exist in '''Top'''. In fact, the forgetful functor ''U'' : '''Top''' → '''Set''' uniquely lifts both limits and colimits and preserves them as well. Therefore, (co)limits in '''Top''' are given by placing topologies on the corresponding (co)limits in '''Set'''.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


Specifically, if ''F'' is a [[diagram (category theory)|diagram]] in '''Top''' and  (''L'', φ) is a limit of ''UF'' in '''Set''', the corresponding limit of ''F'' in '''Top''' is obtained by placing the [[initial topology]] on (''L'', φ). Dually, colimits in '''Top''' are obtained by placing the [[final topology]] on the corresponding colimits in '''Set'''.


Unlike many algebraic categories, the forgetful functor ''U'' : '''Top''' → '''Set''' does not create or reflect limits since there will typically be non-universal [[cone (category theory)|cones]] in '''Top''' covering universal cones in '''Set'''.
* 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.


Examples of limits and colimits in '''Top''' include:
==Test pages ==


*The [[empty set]] (considered as a topological space) is the [[initial object]] of '''Top'''; any [[singleton (mathematics)|singleton]] topological space is a [[terminal object]]. There are thus no [[zero object]]s in '''Top'''.
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*The [[product (category theory)|product]] in '''Top''' is given by the [[product topology]] on the [[Cartesian product]]. The [[coproduct (category theory)|coproduct]] is given by the [[disjoint union (topology)|disjoint union]] of topological spaces.
*[[Displaystyle]]
*The [[equaliser (mathematics)#In category theory|equalizer]] of a pair of morphisms is given by placing the [[subspace topology]] on the set-theoretic equalizer. Dually, the [[coequalizer]] is given by placing the [[quotient topology]] on the set-theoretic coequalizer.
*[[MathAxisAlignment]]
*[[Direct limit]]s and [[inverse limit]]s are the set-theoretic limits with the [[final topology]] and [[initial topology]] respectively.
*[[Styling]]
*[[Adjunction space]]s are an example of [[pushout (category theory)|pushouts]] in '''Top'''.
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


==Other properties==
*[[Inputtypes|Inputtypes (private Wikis only)]]
*The [[monomorphism]]s in '''Top''' are the [[injective]] continuous maps, the [[epimorphism]]s are the [[surjective]] continuous maps, and the [[isomorphism]]s are the [[homeomorphism]]s.
*[[Url2Image|Url2Image (private Wikis only)]]
*The extremal monomorphisms are (up to isomorphism) the [[subspace topology|subspace]] embeddings. Every extremal monomorphism is [[regular morphism (topology)|regular]].
==Bug reporting==
*The extremal epimorphisms are (essentially) the [[quotient map]]s. Every extremal epimorphism is regular.
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 split monomorphisms are (essentially) the inclusions of [[retract]]s into their ambient space.
*The split epimorphisms are (up to isomorphism) the continuous surjective maps of a space onto one of its retracts.
*There are no [[zero morphism]]s in '''Top''', and in particular the category is not [[preadditive category|preadditive]].
*'''Top''' is not [[cartesian closed category|cartesian closed]] (and therefore also not a [[topos]]) since it does not have [[exponential object]]s for all spaces.
 
==Relationships to other categories==
 
*The category of [[pointed topological space]]s '''Top'''<sub>•</sub> is a [[coslice category]] over '''Top'''.  
* The [[homotopy category of topological spaces|homotopy category]] '''hTop''' has topological spaces for objects and [[homotopy equivalent|homotopy equivalence classes]] of continuous maps for morphisms. This is a [[quotient category]] of '''Top'''. One can likewise form the pointed homotopy category '''hTop'''<sub>•</sub>.
*'''Top''' contains the important category '''Haus''' of topological spaces with the [[Hausdorff space|Hausdorff]] property as a [[full subcategory]].  The added structure of this subcategory allows for more epimorphisms:  in fact, the epimorphisms in this subcategory are precisely those morphisms with [[dense set|dense]] [[image (mathematics)|images]] in their [[codomain]]s, so that epimorphisms need not be [[surjective]].
 
== References ==
 
* Herrlich, Horst: ''Topologische Reflexionen und Coreflexionen''. Springer Lecture Notes in Mathematics 78 (1968).
 
* Herrlich, Horst: ''Categorical topology 1971 - 1981''. In: General Topology and its Relations to Modern Analysis and Algebra 5, Heldermann Verlag 1983, pp.&nbsp;279 – 383.
 
* Herrlich, Horst & Strecker, George E.: Categorical Topology - its origins, as examplified by the unfolding of the theory of topological reflections and coreflections before 1971. In: Handbook of the History of General Topology (eds. C.E.Aull & R. Lowen), Kluwer Acad. Publ. vol 1 (1997) pp.&nbsp;255 – 341.
 
* Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). [http://katmat.math.uni-bremen.de/acc/acc.pdf ''Abstract and Concrete Categories''] (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).
 
[[Category:Category-theoretic categories|Topological spaces]]
[[Category:General topology]]

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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