580 -Cardinal arithmetic (5)

February 13, 2009

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 $\alpha$ be a limit ordinal of uncountable cofinality. The set $S\subseteq\alpha$ is stationary in $\alpha$ iff $S\cap C\ne\emptyset$ for all club sets $C\subseteq\alpha.$

For example, let $\lambda$ be a regular cardinal strictly smaller than ${\rm cf}(\alpha).$ Then $S^\alpha_\lambda:=\{\beta<\alpha : {\rm cf}(\beta)=\lambda\}$ is a stationary set, since it contains the $\lambda$-th member of the increasing enumeration of any club in $\alpha.$ This shows that whenever ${\rm cf}(\alpha)>\omega_1,$ there are disjoint stationary subsets of $\alpha.$ Below, we show a stronger result. The notion of stationarity is central to most of set theoretic combinatorics.

Fact 11. Let $S$ be stationary in $\alpha.$

1. $S$ is unbounded in $\alpha.$
2. Let $C$ be club in $\alpha.$ Then $S\cap C$ is stationary.

Proof. 1. $S$ must meet $\kappa\setminus\alpha$ for all $\alpha$ and is therefore unbounded.

2. Given any club sets $C$ and $D,$ $(S\cap C)\cap D=S\cap(C\cap D)\ne\emptyset,$ and it follows that $S\cap C$ is stationary. ${\sf QED}$