Stationary set
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 formula_1 is a cardinal of uncountable cofinality, formula_2 and formula_3 intersects every club set in formula_4 then formula_3 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 formula_3 is a stationary set and formula_7 is a club set, then their intersection formula_8 is also stationary. This is because if formula_9 is any club set, then formula_10 is a club set, thus formula_11 is non empty. Therefore, formula_12 must be stationary. "See also": Fodor's lemma The restriction to uncountable cofinality is in order to avoid trivialities: Suppose formula_1 has countable cofinality. Then formula_14 is stationary in  formula_1 if and only if formula_16 is bounded in formula_1. In particular, if the cofinality of formula_1 is formula_19, then any two stationary subsets of formula_1 have stationary intersection. This is no longer the case if the cofinality of formula_1 is uncountable. In fact, suppose formula_1 is moreover regular and formula_14 is stationary. Then formula_3 can be partitioned into formula_1 many disjoint stationary sets. This result is due to Solovay. If formula_1 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 formula_27, every stationary subset of formula_28 contains a closed subset of order type formula_27. Jech's notion. There is also a notion of stationary subset of formula_30, for formula_31 a cardinal and formula_32 a set such that formula_33, where formula_30 is the set of subsets of formula_32 of cardinality formula_31: formula_37. This notion is due to Thomas Jech. As before, formula_38 is stationary if and only if it meets every club, where a club subset of formula_30 is a set unbounded under formula_40 and closed under union of chains of length at most formula_31. These  notions are in general different, although for formula_42 and formula_43 they coincide in the sense that formula_44 is stationary if and only if formula_45 is stationary in formula_28. 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 Magidor, Foreman and Shelah and has also been used prominently by Woodin. Now let formula_32 be a nonempty set. A set formula_48 is club (closed and unbounded) if and only if there is a function  formula_49 such that formula_50. Here, formula_51 is the collection of finite subsets of formula_52. formula_53 is stationary in formula_54 if and only if it meets every club subset of formula_54. To see the connection with model theory, notice that if formula_56 is a Structure (mathematical logic)|structure with Universe (mathematics)|universe formula_32 in a countable language and formula_58 is a Skolem function for formula_56, then a stationary formula_3 must contain an elementary substructure of formula_56. In fact, formula_53 is stationary if and only if for any such structure formula_56 there is an elementary substructure of formula_56 that belongs to formula_3. External links . Category:Set theory Category:Ordinal numbers