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| {{DISPLAYTITLE:G<sub>δ</sub> set}}
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| 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==
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| 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]].
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| ==Examples==
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| * 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'''.
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| * The set of rational numbers '''Q''' is '''not''' a G<sub>δ</sub> set in '''R'''. 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]].
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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]].
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| A more elaborate example of a G<sub>δ</sub> set is given by the following theorem:
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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> contains a dense 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>
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| ==Properties==
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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>
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| # 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.
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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.
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| ===Basic properties===
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| * The [[complement (set theory)|complement]] of a G<sub>δ</sub> set is an [[Fσ set|F<sub>σ</sub>]] set.
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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.
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| * 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.
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| * A [[topological subspace|subspace]] ''A'' of a [[completely metrizable]] space ''X'' is itself completely metrizable 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>). Then <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==
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| A '''[[Gδ space|G<sub>δ</sub> space]]''' is a topological space in which every [[closed set]] is a G<sub>δ</sub> set {{harv|Johnson|1970}}. 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==
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| * [[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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| * [[P-space|''P''-space]], any space having the property that every G<sub>δ</sub> set is open
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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:G Set}}
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| [[Category:General topology]]
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| [[Category:Descriptive set theory]]
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