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  is a cardinal of uncountable cofinality,  and  intersects every club set in  then  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  is a stationary set and  is a club set, then their intersection  is also stationary. This is because if  is any club set, then  is a club set, thus  is nonempty. Therefore,  must be stationary.

See also: Fodor's lemma

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

This is no longer the case if the cofinality of  is uncountable. In fact, suppose  is moreover regular and  is stationary. Then  can be partitioned into  many disjoint stationary sets. This result is due to Solovay. If  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 , every stationary subset of  contains a closed subset of order type .

Jech's notion
There is also a notion of stationary subset of , for  a cardinal and  a set such that , where  is the set of subsets of  of cardinality : . This notion is due to Thomas Jech. As before,  is stationary if and only if it meets every club, where a club subset of  is a set unbounded under  and closed under union of chains of length at most . These notions are in general different, although for  and  they coincide in the sense that  is stationary if and only if  is stationary in .

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  be a nonempty set. A set  is club (closed and unbounded) if and only if there is a function  such that . Here,  is the collection of finite subsets of .

 is stationary in  if and only if it meets every club subset of .

To see the connection with model theory, notice that if  is a structure with universe  in a countable language and  is a Skolem function for , then a stationary  must contain an elementary substructure of . In fact,  is stationary if and only if for any such structure  there is an elementary substructure of  that belongs to .

References 

 Foreman, Matthew (2002) 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. pp. 73–94.  File at

External links 
 

Set theory
Ordinal numbers