, particularly in set theory
and model theory
, there are at least three notions of stationary set
is a cardinal
of uncountable cofinality
, and intersects
is called a stationary set
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 are club sets we have: . Now is a club set as it is the intersection of two club sets. So is non empty. But then must be stationary as is arbitrary.
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 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.
There is also a notion of stationary subset of
a cardinal and
a set such that
. 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
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.
There is yet a third notion, model theoretic in nature and sometimes referred to as generalized
stationarity. This notion is probably due to Magidor
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