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{{Redirect|Thin set (set theory)|other uses|Thin set (disambiguation){{!}}Thin set}}
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In [[mathematics]], particularly in [[set theory]] and [[model theory]], there are at least three notions of '''stationary set''':
 
==Classical notion==
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]].
 
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.
 
''See also'': [[Fodor's lemma]]
 
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.
 
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'''.
 
==Jech's notion==
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>.
 
The appropriate version of Fodor's lemma also holds for this notion.
 
==Generalized notion==
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]].
 
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>.
 
<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>.  
 
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>.
 
== References ==
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.&nbsp;73–94
File at [http://www.math.uci.edu/sub2/Foreman/homepage/hajfin.ps]
 
== External links ==
* {{planetmath reference |id=3228|title=Stationary set}}
 
[[Category:Set theory]]
[[Category:Ordinal numbers]]

Revision as of 12:58, 9 February 2014

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