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.