At the end of last lecture we defined club sets and showed that the diagonal intersection of club subsets of a regular cardinal is club.
Definition 10. Let be a limit ordinal of uncountable cofinality. The set is stationary in iff for all club sets
For example, let be a regular cardinal strictly smaller than Then is a stationary set, since it contains the -th member of the increasing enumeration of any club in This shows that whenever there are disjoint stationary subsets of Below, we show a stronger result. The notion of stationarity is central to most of set theoretic combinatorics.
Fact 11. Let be stationary in
- is unbounded in
- Let be club in Then is stationary.
Proof. 1. must meet for all and is therefore unbounded.
2. Given any club sets and and it follows that is stationary.