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| {{DISPLAYTITLE:G<sub>δ</sub> set}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
| In the mathematical field of [[topology]], a '''G<sub>δ</sub> set''' is a [[subset]] of a [[topological space]] that is a countable intersection of open sets. The notation originated in [[Germany]] with ''G'' for ''[[wikt:Gebiet#German|Gebiet]]'' (''[[German language|German]]'': area, or neighborhood) meaning [[open set]] in this case and δ for ''[[wikt:Durchschnitt#German|Durchschnitt]]'' (''German'': [[intersection (set theory)|intersection]]).
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| The term '''inner limiting set''' is also used. G<sub>δ</sub> sets, and their dual [[F-sigma set|F<sub>σ</sub> sets]], are the second level of the [[Borel hierarchy]].
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| ==Definition==
| | 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]] |
| In a topological space a '''G<sub>δ</sub> set''' is a [[countable]] [[intersection (set theory)|intersection]] of [[open set]]s. The G<sub>δ</sub> sets are exactly the level <math>\mathbf{\Pi}^0_2</math> sets of the [[Borel hierarchy]].
| | * Only registered users will be able to execute this rendering mode. |
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| ==Examples==
| | Registered users will be able to choose between the following three rendering modes: |
| * Any open set is trivially a G<sub>δ</sub> set
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| * The [[irrational numbers]] are a G<sub>δ</sub> set in '''R''', the real numbers, as they can be written as the intersection over all [[rational number|rational]] numbers ''q'' of the [[complement (set theory)|complement]] of {''q''} in '''R'''. Note that the set of [[Rational number|rational]] numbers is not a G<sub>δ</sub> set in '''R'''.
| | '''MathML''' |
| | :<math forcemathmode="mathml">E=mc^2</math> |
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| * The rational numbers '''Q''' are '''not''' a G<sub>δ</sub> set. If we were able to write '''Q''' as the intersection of open sets ''A<sub>n</sub>'', each ''A<sub>n</sub>'' would have to be [[dense set|dense]] in '''R''' since '''Q''' is dense in '''R'''. However, the construction above gave the irrational numbers as a countable intersection of open dense subsets. Taking the intersection of both of these sets gives the [[empty set]] as a countable intersection of open dense sets in '''R''', a violation of the [[Baire category theorem]].
| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
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| * The zero-set of a derivative of an everywhere differentiable real-valued function on '''R''' is a G<sub>δ</sub> set; it can be a dense set with empty interior, as shown by [[Pompeiu derivative#Pompeiu's construction|Pompeiu's construction]].
| | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| A more elaborate example of a G<sub>δ</sub> set is given by the following theorem:
| | <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]. |
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| '''Theorem:''' The set <math>D=\left\{f \in C([0,1]) : f \text{ is not differentiable at any point of } [0,1] \right\}</math> is dense in <math>C([0,1])</math> and contains a G<sub>δ</sub> subset of the metric space <math>C([0,1])</math><ref name="Negrepontis 1997">{{cite book|last1=Νεγρεπόντης|first1=Σ.|last2=Ζαχαριάδης|first2=Θ.|last3=Καλαμίδας|first3=Ν.|last4=Φαρμάκη|first4=Β.|title=Γενική Τοπολογία και Συναρτησιακη Ανάλυσγη|year=1997|publisher=Εκδόσεις Συμμετρία|location=Αθήνα, Ελλάδα|isbn=960-266-178-Χ|pages=55–64|url=http://www.simmetria.gr/eshop/?149,%CD%C5%C3%D1%C5%D0%CF%CD%D4%C7%D3-%D3.-%C6%C1%D7%C1%D1%C9%C1%C4%C7%D3-%C8.-%CA%C1%CB%C1%CC%C9%C4%C1%D3-%CD.-%D6%C1%D1%CC%C1%CA%C7-%C2.-%C3%E5%ED%E9%EA%DE-%D4%EF%F0%EF%EB%EF%E3%DF%E1-%EA%E1%E9-%D3%F5%ED%E1%F1%F4%E7%F3%E9%E1%EA%DE-%C1%ED%DC%EB%F5%F3%E7|accessdate=3 April 2011|language=Greek|chapter=2, Πλήρεις Μετρικοί Χώροι}}</ref>
| | ==Demos== |
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| ==Properties== | | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| The notion of G<sub>δ</sub> sets in [[Metric space|metric]] (and [[Topological space|topological]]) spaces is strongly related to the notion of [[Complete metric space|completeness]] of the metric space as well as to the [[Baire category theorem]]. This is described by the [[Mazurkiewicz]] theorem:
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| '''Theorem''' ([[Mazurkiewicz]]): Let <math>(\mathcal{X},\rho)</math> be a complete metric space and <math>A\subset\mathcal{X}</math>. Then the following are equivalent:
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| # <math>A</math> is a G<sub>δ</sub> subset of <math>\mathcal{X}</math>
| | ** 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]] |
| # There is a [[Metric (mathematics)|metric]] <math>\sigma</math> on <math>A</math> which is [[Metric_(mathematics)#Equivalence_of_metrics|equivalent]] to <math>\rho | A</math> such that <math>(A,\sigma)</math> is a complete metric space.
| | ** [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. |
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| A key property of <math>G_\delta</math> sets is that they are the possible sets at which a function from a topological space to a metric space is [[continuous function|continuous]]. Formally: The set of points where a function <math>f</math> is continuous is a <math>G_\delta</math> set. This is because continuity at a point <math>p</math> can be defined by a <math>\Pi^0_2</math> formula, namely: For all positive integers <math>n</math>, there is an open set <math>U</math> containing <math>p</math> such that <math>d(f(x),f(y)) < 1/n</math> for all <math>x, y</math> in <math>U</math>. If a value of <math>n</math> is fixed, the set of <math>p</math> for which there is such a corresponding open <math>U</math> is itself an open set (being a union of open sets), and the [[universal quantifier]] on <math>n</math> corresponds to the (countable) intersection of these sets. In the real line, the converse holds as well; for any G<sub>δ</sub> subset ''A'' of the real line, there is a function ''f'': '''R''' → '''R''' which is continuous exactly at the points in ''A''. As a consequence, while it is possible for the irrationals to be the set of continuity points of a function (see the [[popcorn function]]), it is impossible to construct a function which is continuous only on the rational numbers.
| | ==Test pages == |
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| ===Basic properties===
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
| * The [[complement (set theory)|complement]] of a G<sub>δ</sub> set is an [[Fσ set|F<sub>σ</sub>]] set. | | *[[Displaystyle]] |
| | *[[MathAxisAlignment]] |
| | *[[Styling]] |
| | *[[Linebreaking]] |
| | *[[Unique Ids]] |
| | *[[Help:Formula]] |
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| * The intersection of countably many G<sub>δ</sub> sets is a G<sub>δ</sub> set, and the union of ''finitely'' many G<sub>δ</sub> sets is a G<sub>δ</sub> set; a countable union of G<sub>δ</sub> sets is called a G<sub>δσ</sub> set. | | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| * In [[metrizable]] spaces, every [[closed set]] is a G<sub>δ</sub> set and, dually, every open set is an F<sub>σ</sub> set.
| | ==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 . |
| * A [[topological subspace|subspace]] ''A'' of a [[topologically complete]] space ''X'' is itself topologically complete if and only if ''A'' is a G<sub>δ</sub> set in ''X''.
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| * A set that contains the intersection of a countable collection of [[dense set|dense]] open sets is called '''[[comeagre set|comeagre]]''' or '''residual.''' These sets are used to define [[generic property|generic properties]] of topological spaces of functions.
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| The following results regard [[Polish space]]s<ref name="Fremlin 2003">{{cite book|last=Fremlin|first=D.H.|title=Measure Theory, Volume 4|year=2003|publisher=Digital Books Logistics|location=Petersburg, England|isbn=0-9538129-4-4|pages=334–335|url=http://www.essex.ac.uk/maths/people/fremlin/mt.htm|accessdate=1 April 2011|chapter=4, General Topology}}</ref>:
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| * Let <math>(\mathcal{X},\mathcal{T})</math> be a [[Polish space|Polish topological space]] and let <math>G\subset\mathcal{X}</math> be a G<sub>δ</sub> set (with respect to <math>\mathcal{T}</math>). The <math>G</math> is a Polish space with respect to the [[subspace topology]] on it. | |
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| * Topological characterization of Polish spaces: If <math>\mathcal{X}</math> is a [[Polish space]] then it is [[Homeomorphism|homeomorphic]] to a G<sub>δ</sub> subset of a [[Compact space|compact]] [[metric space]].
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| ==G<sub>δ</sub> space== | |
| A [[Gδ space|'''G<sub>δ</sub> space''']] is a topological space in which every [[closed set]] is a G<sub>δ</sub> set ([http://www.jstor.org/stable/2317335 Johnson, 1970]).{{Citation needed|date=August 2008}} A [[normal space]] which is also a G<sub>δ</sub> space is '''[[perfectly normal space|perfectly normal]]'''. Every metrizable space is perfectly normal, and every perfectly normal space is [[completely normal]]: neither implication is reversible.
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| ==See also== | |
| * [[Fσ set|F<sub>σ</sub> set]], the [[duality (mathematics)|dual]] concept; note that "G" is German (''[[wikt:Gebiet#German|Gebiet]]'') and "F" is French (''[[wikt:fermé#French|fermé]]'').
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| ==References==
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| * [[John L. Kelley]], ''General topology'', [[Van Nostrand Reinhold|van Nostrand]], 1955. P.134.
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| * {{Cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}} P. 162.
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| * {{Cite book | last=Fremlin | first=D.H. | title=Measure Theory, Volume 4 | origyear=2003 | publisher=Digital Books Logostics | location=Petersburg, England | isbn=0-9538129-4-4 | year=2003 | url=http://www.essex.ac.uk/maths/people/fremlin/mt.htm|accessdate=1 April 2011|chapter=4, General Topology | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}} P. 334.
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| * Roy A. Johnson (1970). "A Compact Non-Metrizable Space Such That Every Closed Subset is a G-Delta". ''The American Mathematical Monthly'', Vol. 77, No. 2, pp. 172–176. [http://www.jstor.org/stable/2317335 on JStor]
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| ==Notes==
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| <references />
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| {{DEFAULTSORT:Gδ Set}}
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| [[Category:General topology]]
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| [[Category:Descriptive set theory]]
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| [[ja:Gδ集合]]
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| [[pl:Zbiór typu G-delta]]
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