Stationary subset

Stationary set

In mathematics, particularly in set theory and model theory, there are at least three notions of stationary set:

Classical notion

If kappa is a cardinal of uncountable cofinality, Csubseteqkappa, and C intersects every club in kappa, then C is called a stationary set. If C is not stationary then it is a thin set.

In fact the intersection of a stationary set and a club set is itself stationary. This is true because if S is stationary and C_1 , C_2 are club sets we have: S cap (C_1 cap C_2) = (S cap C_1) cap C_2. Now C_1 cap C_2 is a club set as it is the intersection of two club sets. So S cap (C_1 cap C_2) is non empty. But then (S cap C_1) must be stationary as C_2 is arbitrary.

See also: Fodor's lemma

The restriction to uncountable cofinality is in order to avoid trivialities: Suppose kappa has countable cofinality. Then Ssubsetkappa is stationary in kappa if and only if kappasetminus S is bounded in kappa. In particular, if the cofinality of kappa is omega=aleph_0, then any two stationary subsets of kappa have stationary intersection.

This is no longer the case if the cofinality of kappa is uncountable. In fact, suppose kappa is regular and Ssubsetkappa is stationary. Then S can be partitioned into kappa many disjoint stationary sets. This result is due to Solovay. If kappa is a successor cardinal, this result is due to 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 [X]^lambda, for lambda a cardinal and X a set such that |X|gelambda, where [X]^lambda={Ysubset X:|Y|=lambda}. This notion is due to Thomas Jech. As before, Ssubset[X]^lambda is stationary if and only if it meets every club, where a club subset of [X]^lambda is a set unbounded under subset and closed under union of chains of length at most lambda. These notions are in general different, although for X=omega_1 and lambda=aleph_0 they coincide in the sense that Ssubset[omega_1]^omega is stationary if and only if Scapomega_1 is stationary in omega_1.

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 X be a nonempty set. A set Csubset{mathcal P}(X) is club (closed and unbounded) if and only if there is a function F:[X]^{ such that C={z:F[[z]^{. Here, [y]^{ is the collection of finite subsets of y.

Ssubset{mathcal P}(X) is stationary in {mathcal P}(X) if and only if it meets every club subset of {mathcal P}(X).

To see the connection with model theory, notice that if M is a structure with universe X in a countable language and F is a Skolem function for M, then a stationary S must contain an elementary substructure of M. In fact, Ssubset{mathcal P}(X) is stationary if and only if for any such structure M there is an elementary substructure of M that belongs to S.


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 File at

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