Stationary set

Remarks

Example: each club set $D$ in $\kappa$ is stationary. Check: for each club $C$, the intersection $C \cap D$ is club, hence unbounded, hence nonempty. Example: the set $\set{\alpha < \omega_2 : \operatorname{cf}(\alpha) = \omega}$ is stationary in $\omega_2$. Check: each club $C$ in $\omega_2$ contains the supremum of its first $\omega$ elements, and this supremum has cofinality $\omega$. Non-example: a bounded set $S \subseteq \kappa$, with $S \subseteq \beta$ for some $\beta < \kappa$, misses the club set $\set{\alpha : \beta \le \alpha < \kappa}$. The sets that are not stationary form a $\kappa$-complete ideal, the nonstationary ideal; it is closed under diagonal unions. The same definition is used for a limit ordinal of uncountable cofinality in place of $\kappa$. By a theorem of Solovay, each stationary subset of $\kappa$ is the union of $\kappa$ pairwise disjoint stationary sets.