In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset.
Classical notion If κ {\displaystyle \kappa } is a cardinal of uncountable cofinality, S ⊆ κ , {\displaystyle S\subseteq \kappa ,} and S {\displaystyle S} intersects every club set in κ , {\displaystyle \kappa ,} then S {\displaystyle S} 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 in number theory. If S {\displaystyle S} is a stationary set and C {\displaystyle C} is a club set, then their intersection S ∩ C {\displaystyle S\cap C} is also stationary. This is because if D {\displaystyle D} is any club set, then C ∩ D {\displaystyle C\cap D} is a club set, thus ( S ∩ C ) ∩ D = S ∩ ( C ∩ D ) {\displaystyle (S\cap C)\cap D=S\cap (C\cap D)} is nonempty. Therefore, ( S ∩ C ) {\displaystyle (S\cap C)} must be stationary. See also: Fodor's lemma The restriction to uncountable cofinality is in order to avoid trivialities: Suppose κ {\displaystyle \kappa } has countable cofinality. Then S ⊆ κ {\displaystyle S\subseteq \kappa } is stationary in κ {\displaystyle \kappa } if and only if κ ∖ S {\displaystyle \kappa \setminus S} is bounded in κ {\displaystyle \kappa } . In particular, if the cofinality of κ {\displaystyle \kappa } is ω = ℵ 0 {\displaystyle \omega =\aleph _{0}} , then any two stationary subsets of κ {\displaystyle \kappa } have stationary intersection. This is no longer the case if the cofinality of κ {\displaystyle \kappa } is uncountable. In fact, suppose κ {\displaystyle \kappa } is moreover regular and S ⊆ κ {\displaystyle S\subseteq \kappa } is stationary. Then S {\displaystyle S} can be partitioned into κ {\displaystyle \kappa } many disjoint stationary sets. This result is due to Solovay. If κ {\displaystyle \kappa } is a successor cardinal, this result is due to Ulam and is easily shown by means of what is called an Ulam matrix. H. Friedman has shown that for every countable successor ordinal β {\displaystyle \beta } , every stationary subset of ω 1 {\displaystyle \omega _{1}} contains a closed subset of order type β {\displaystyle \beta } .
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