Club Sets and Stationary Sets

Club and stationary sets are fundamental tools in combinatorial set theory, providing a framework for studying "large" subsets of ordinals and their intersection properties.

Closed Unbounded Sets

Definition7.5Club set

Let $\kappa$ be an uncountable regular cardinal. A set $C \subseteq \kappa$ is closed unbounded (club) if:

Closed: If $\alpha < \kappa$ is a limit point of $C$ (i.e., $\sup(C \cap \alpha) = \alpha$ ), then $\alpha \in C$ .
Unbounded: $\sup C = \kappa$ , i.e., for every $\alpha < \kappa$ , there exists $\beta \in C$ with $\beta > \alpha$ .

ExampleExamples of club sets

The set of all limit ordinals less than $\omega_1$ is club in $\omega_1$ .
For any function $f: \kappa \to \kappa$ , the set of closure points $C_f = \{\alpha < \kappa : f[\alpha] \subseteq \alpha\}$ is club (assuming $\kappa$ regular).
The intersection of any two club sets is club.
The intersection of fewer than $\kappa$ club sets in $\kappa$ is club (when $\kappa$ is regular).

Stationary Sets

Definition7.6Stationary set

A set $S \subseteq \kappa$ is stationary if $S \cap C \neq \emptyset$ for every club $C \subseteq \kappa$ . Equivalently, $S$ is stationary if it is not disjoint from any club set, i.e., $S$ is "too large to avoid."

Theorem7.3Fodor's theorem (pressing-down lemma)

Let $\kappa$ be a regular uncountable cardinal and $S \subseteq \kappa$ a stationary set. If $f: S \to \kappa$ is a regressive function (i.e., $f(\alpha) < \alpha$ for all $\alpha \in S$ with $\alpha > 0$ ), then $f$ is constant on a stationary set: there exists $\gamma < \kappa$ such that $f^{-1}(\gamma)$ is stationary.

Applications

ExampleApplications of stationary sets

The set $S^\kappa_\omega = \{\alpha < \kappa : \mathrm{cf}(\alpha) = \omega\}$ is stationary in $\kappa$ for any regular $\kappa > \omega$ . This is because the club filter cannot "avoid" cofinality $\omega$ ordinals.
Ulam matrices: Using stationary sets, one can show that $\omega_1$ does not carry a countably additive two-valued measure, proving $\omega_1$ is not measurable.
Diamond principle ( $\Diamond$ ): A combinatorial principle (consistent with and implied by V = L) stating that there exists a sequence $\langle A_\alpha : \alpha < \omega_1 \rangle$ that "guesses" every subset of $\omega_1$ on a stationary set.

RemarkThe club filter

The collection of club subsets of a regular uncountable cardinal $\kappa$ generates a normal filter called the club filter. Its dual ideal consists of the nonstationary sets. The stationary sets are precisely those that are positive with respect to this filter (i.e., not in the ideal). The quotient algebra $\mathcal{P}(\kappa)/\mathrm{NS}_\kappa$ (power set modulo nonstationary ideal) is a fundamental object in set theory, and questions about its saturation and precipitousness connect to large cardinal hypotheses.