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| {{Redirect|Thin set (set theory)|other uses|Thin set (disambiguation){{!}}Thin set}}
| | I'm Janice and I live with my husband and our three children in Koln Innenstadt, in the NW south area. My hobbies are Sailing, Musical instruments and Urban exploration.<br><br>my site ... [http://Www.businessvibes.com/gail-plastasia-2013/?q=content/fifa-15-coin-hack-0 FIFA Coin Generator] |
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| In [[mathematics]], particularly in [[set theory]] and [[model theory]], there are at least three notions of '''stationary set''':
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| ==Classical notion==
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| If <math> \kappa \,</math> is a [[cardinal number|cardinal]] of uncountable [[cofinality]], <math> S \subseteq \kappa \,,</math> and <math> S \,</math> [[Intersection (set theory)|intersects]] every [[club set]] in <math> \kappa \,,</math> then <math> S \,</math> is called a '''stationary set'''. If a set is not stationary, then it is called a '''thin set'''. This notion should not be confused with the notion of a [[Thin set (Serre)|thin set in number theory]].
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| If <math> S \,</math> is a stationary set and <math> C \,</math> is a club set, then their intersection <math> S \cap C \,</math> is also stationary. Because if <math> D \,</math> is any club set, then <math> C \cap D \,</math> is a club set because the intersection of two club sets is club. Thus <math> (S \cap C) \cap D = S \cap (C \cap D) \,</math> is non empty. Therefore <math> (S \cap C) \,</math> must be stationary.
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| ''See also'': [[Fodor's lemma]]
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| The restriction to uncountable cofinality is in order to avoid trivialities: Suppose <math>\kappa</math> has countable cofinality. Then <math>S\subset\kappa</math> is stationary in <math>\kappa</math> if and only if <math>\kappa\setminus S</math> is bounded in <math>\kappa</math>. In particular, if the cofinality of <math>\kappa</math> is <math>\omega=\aleph_0</math>, then any two stationary subsets of <math>\kappa</math> have stationary intersection.
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| This is no longer the case if the cofinality of <math>\kappa</math> is uncountable. In fact, suppose <math>\kappa</math> is [[regular cardinal|regular]] and <math>S\subset\kappa</math> is stationary. Then <math>S</math> can be partitioned into <math>\kappa</math> many disjoint stationary sets. This result is due to [[Robert M. Solovay|Solovay]]. If <math>\kappa</math> is a [[successor cardinal]], this result is due to [[Stanislaw Ulam|Ulam]] and is easily shown by means of what is called an '''Ulam matrix'''.
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| ==Jech's notion==
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| There is also a notion of stationary subset of <math>[X]^\lambda</math>, for <math>\lambda</math> a cardinal and <math>X</math> a set such that <math>|X|\ge\lambda</math>, where <math>[X]^\lambda</math> is the set of subsets of <math>X</math> of cardinality <math>\lambda</math>: <math>[X]^\lambda=\{Y\subset X:|Y|=\lambda\}</math>. This notion is due to [[Thomas Jech]]. As before, <math>S\subset[X]^\lambda</math> is stationary if and only if it meets every club, where a club subset of <math>[X]^\lambda</math> is a set unbounded under <math>\subset</math> and closed under union of chains of length at most <math>\lambda</math>. These notions are in general different, although for <math>X=\omega_1</math> and <math>\lambda=\aleph_0</math> they coincide in the sense that <math>S\subset[\omega_1]^\omega</math> is stationary if and only if <math>S\cap\omega_1</math> is stationary in <math>\omega_1</math>.
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| The appropriate version of Fodor's lemma also holds for this notion.
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| ==Generalized notion==
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| There is yet a third notion, model theoretic in nature and sometimes referred to as '''generalized''' stationarity. This notion is probably due to [[Menachem Magidor|Magidor]], [[Matthew Foreman|Foreman]] and [[Saharon Shelah|Shelah]] and has also been used prominently by [[W. Hugh Woodin|Woodin]].
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| Now let <math>X</math> be a nonempty set. A set <math>C\subset{\mathcal P}(X)</math> is club (closed and unbounded) if and only if there is a function <math>F:[X]^{<\omega}\to X</math> such that <math>C=\{z:F[[z]^{<\omega}]\subset z\}</math>. Here, <math>[y]^{<\omega}</math> is the collection of finite subsets of <math>y</math>.
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| <math>S\subset{\mathcal P}(X)</math> is stationary in <math>{\mathcal P}(X)</math> if and only if it meets every club subset of <math>{\mathcal P}(X)</math>.
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| To see the connection with model theory, notice that if <math>M</math> is a [[Structure (mathematical logic)|structure]] with [[Universe (mathematics)|universe]] <math>X</math> in a countable language and <math>F</math> is a [[Skolem function]] for <math>M</math>, then a stationary <math>S</math> must contain an elementary substructure of <math>M</math>. In fact, <math>S\subset{\mathcal P}(X)</math> is stationary if and only if for any such structure <math>M</math> there is an elementary substructure of <math>M</math> that belongs to <math>S</math>.
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| == References ==
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| Matthew Foreman, ''Stationary sets, Chang's Conjecture and partition theory'', in Set Theory (The Hajnal Conference) DIMACS Ser. Discrete Math. Theoret. Comp. Sci., 58, Amer. Math. Soc., Providence, RI. 2002 pp. 73–94
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| File at [http://www.math.uci.edu/sub2/Foreman/homepage/hajfin.ps]
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| == External links ==
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| * {{planetmath reference |id=3228|title=Stationary set}}
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| [[Category:Set theory]]
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| [[Category:Ordinal numbers]]
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I'm Janice and I live with my husband and our three children in Koln Innenstadt, in the NW south area. My hobbies are Sailing, Musical instruments and Urban exploration.
my site ... FIFA Coin Generator