Closed and unbounded subsets of ordinals, more commonly referred to as *club* sets, play a prominent role in modern set theory. We intuitively think of clubs as the "large" subsets of $\kappa$ and the stationary subsets as the "not small" subsets of $\kappa$, though this is sort of a boring way to look at them. They arise from considering the natural topology on the class of ordinals and often exhibit substantial reflection properties. # Basic notion - subsets of ordinals Given an ordinal $\kappa$, the basic open intervals are pairs of ordinals $(\alpha, \beta)=\{\gamma : \alpha <\gamma < \beta\}$ where $\beta <\kappa$. Closed intervals are defined similarly, so closed intervals are topologically closed. Considering a typical example of an interval of ordinals $[\lambda, \lambda+1, \lambda+2, \dots)$ it appears there are *more* successor ordinals than limits, but club (and also stationary) sets favor limit ordinals in the sense that they concentrate on them. Hence the opposite view-point is more useful when considering club sets, i.e., there are "more" limit ordinals. ## Club sets Although the definition can be applied to all infinite ordinals, we assume $\kappa >\omega$ is a regular cardinal for this and subsequent sections. >[!info] Definition >A set $C\subseteq \kappa$ is *closed unbounded* or *club* in $\kappa$ if and only if $C$ is > - *unbounded* in $\kappa$: for every $\alpha <\kappa$ there is some $\beta \in C$ with occurring above $\alpha$ in the natural ordering; > - *closed*: if $B\subseteq C$ is bounded in $\kappa$ (i.e., there is some $\gamma\in \kappa$ with $\beta\leq \gamma$ for each $\beta\in B$), then $sup(B)\in C$. > Equivalently: if $\lambda < \kappa$ and $\lambda$ a limit with $C\cap\lambda$ unbounded in $\lambda$, then $\lambda\in C$. >[!example] >Typical examples of club sets include > - the collection of limit ordinals below $\kappa$ > - the collection of limits of limit ordinals below $\kappa$ > - all "tails" in $\kappa$: $\{\lambda : \alpha\leq \lambda <\kappa\}$ for each $\alpha <\kappa$. It is fairly straightforward to construct a club subset of $\kappa$. Given a sequence $\langle\gamma_\alpha\rangle_{\alpha < \kappa}$ of ordinals smaller than $\kappa$ arbitrarily pick $\gamma_{\alpha +1}$ also smaller than $\kappa$. At limit stages, take the supremum of the sequence already constructed. It is clear that club subsets of $\kappa$ all have size $\kappa$ and their enumeration functions $f:\kappa\rightarrow\kappa$ are all continuous and increasing. **Lemma.** Let $\kappa$ be an uncountable regular cardinal and $f:\kappa \to \kappa$ . 1. The set $\{\gamma \in \kappa \mid \forall \nu < \gamma\ (f(\nu)<\gamma)\} = \{\gamma\in \kappa \mid f''\gamma \subseteq \gamma\}$ is club in $\kappa$. 2. If $f$ is increasing and continuous then the set $\{\gamma \in \kappa \mid f(\gamma)=\gamma\}$ is club in $\kappa$. ### The club filter The intersection of two club subsets of $\kappa$ is also club in $\kappa$. In fact, given any sequence of fewer than $\kappa$-many club subsets of $\kappa$, their intersection is also club in $\kappa$. Further, any collection of fewer than $\kappa$-many club subsets is also closed under *diagonal* intersections, a fact used in characterizing the stationary subsets of $\kappa$. In particular, the club subsets of $\kappa$ form a [filter](Filter.md "Filter") over $\kappa$. Note that the intersection of $\kappa$-many clubs might be empty, so this filter is not an ultrafilter in general. ## Stationary sets >[!info] Definition >A set $S\subseteq \kappa$ is *stationary in $\kappa$* if $S$ intersects all club subsets of $\kappa$. As mentioned above, one intuitively thinks of the collection of stationary subsets of $\kappa$ as the "not small" subsets of $\kappa$. Several facts about stationary sets are immediate: - All club subsets of $\kappa$ are also stationary in $\kappa$; - the supremum of a stationary subset of $\kappa$, is $\kappa$; - the intersection of a club set with a stationary set is stationary; - if $S$ is a stationary set and also the union of less than $\kappa$-many sets $S_\alpha$, then at least one such set is also stationary, in other words, stationary subsets of $\kappa$ cannot be partitioned into a small number of small sets. >[!example] > For regular $\lambda<\kappa$ the following set is stationary: > $E_{\lambda}^{\kappa}:=\left\{ \alpha<\kappa\mid\mathrm{cf}\left(\alpha\right)=\lambda\right\}$ ### Fodor's Lemma Fodor's Lemma (improving upon Alexandrov-Urysohn, 1929) is the basic, fundamental result concerning the concept of stationarity. Call a function $f:\kappa\to\kappa$ *regressive* if $f(\alpha) < \alpha$ for all non-zero ordinals smaller than $\kappa$. **Fodor's lemma.** If $f$ is a regressive function with domain a stationary subset $S$ of $\kappa$, then there is some stationary subset $S'$ of $S$ on which $f$ is constant. Using Fodor's lemma, Solovay proved that each stationary subset of $\kappa$ can be split into two, in fact $\kappa$-many disjoint stationary sets. Another application of Fodor's lemma is used to prove a result concerning families of sets that are as different as possible, i.e., any two distinct sets in the family have the same intersection. The result is more popularly known as the $\Delta$-system Lemma (originally established by Marczewski): Given a family of finite sets (infinite sets usually require CH), of size $\kappa$ there is a subfamily of size $\kappa$ which forms a $\Delta$-system. # Generalized notions >[!info] Definition >For an arbitrary set $X$, $C\subseteq\mathcal{P}(X)$ is called *club in $\mathcal{P}(X)$* if there is some algebra $\mathfrak{A}=\left\langle X,f_{n}\right\rangle _{n<\omega}$ (where $f_{n}:X^{k}\to X$ for some $k$) such that $C=C_{\mathfrak{A}}:=\left\{ z\in\mathcal{P}(X)\mid\forall n(f_{n}''z^{k}\subseteq z)\right\}$ i.e. the collection of all subsets of $X$ closed under all functions of $\mathfrak{A}$. >$S\subseteq\mathcal{P}(X)$ is called *stationary in $\mathcal{P}(X)$* if for every algebra $\mathfrak{A}$ on $X$, $S\cap C_{\mathfrak{A}}\ne\varnothing$. >[!note] Remark >- The notions of club and stationary sets can also be defined using single functions $F:\left[X\right]^{<\omega}\to X$, where a club is the set of all sets closed under such $F$ and a stationary is such that for each such $F$ there is a member closed under $F$. >- The club subsets generate a filter on $\mathcal{P}(X)$, and the stationary sets are the positives sets with respect to this filter. >- Using Skolem functions, the club filter is also generated by club sets which consist of all *elementary* substructures of some structure on $X$. And similarly - a stationary set is one that contains an elementary substructure of any structure on $X$. >[!info] Definition >For a set $X$ and a cardinal $\lambda$, denote $\mathcal{P}_{\lambda}(X):=\{ y \subseteq X \mid |Y|<\lambda\}$. > The notions of *club/stationary subset of $\mathcal{P}_{\lambda}(X)$* are defined by replacing $\mathcal{P}(X)$ with $\mathcal{P}_{\lambda}(X)$ in the above defintion. >[!info] Alternative definition > A set $C \subset P_\kappa(X)$ is unbounded if for every $x \in P_\kappa(X)$ there exists a $y \supset x$ such that $y \in C$. > A set $C \subset P_\kappa(X)$ is closed if for any chain $x_0 \subset x_1 \subset \ldots \subset x_{\xi} \subset \ldots$, $\xi<\alpha$, of sets in $C$, with $\alpha<\kappa$, the union $\bigcup_{\xi<\alpha} x_{\xi}$ is in $C$. > A set $C \subset P_\kappa(X)$ is closed unbounded if it is closed and unbounded. > A set $S \subset P_\kappa(X)$ is stationary if $S \cap C \neq \emptyset$ for every closed unbounded $C \subset P_\kappa(X)$. ## The club filter and non-stationary ideal >[!info] Definition ^ns-ideal >The **closed unbounded filter** on $P_\kappa(X)$ is the filter generated by the closed unbounded sets. >The dual of the club filter is the *non-stationary ideal*. **Theorem (Jech/Kueker).** The closed unbounded filter on $P_\kappa(X)$ is $\kappa$-complete. >[!info] Definition >The *diagonal intersection* of subsets of $P_\kappa(A)$ is defined as follows > $ > \triangle_{a \in A} X_a=\left\{x \in P_\kappa(A): x \in \bigcap_{a \in x} X_a\right\} > $ **Lemma.** If $\left\{C_a: a \in A\right\}$ is a collection of closed unbounded subsets of $P_\kappa(A)$ then its diagonal intersection is closed unbounded. **Theorem (Jech).** If $f$ is a function on a stationary set $S \subset P_\kappa(\lambda)$ and if $f(x) \in x$ for every nonempty $x \in S$, then there exist a stationary set $T \subset S$ and some $a \in A$ such that $f(x)=a$ for all $a \in T$. >[!note] Remark >This is a generalization of [[Club sets and stationary sets#Fodor's Lemma|Fodor's lemma]]. >[!note] Notation >These two properties are refered to as *normality* of the club filter and the non-stationary ideal. ## Club and stationary subsets of $P_{\omega_{1}}(X)$ An important special case is obtained by looking at *countable* subsets of $X$. In this case, another way of generating the club filter is using sets $C$ which are closed under countable unions and unbounded in the sense that for every $a\in\mathcal{P}_{\omega_{1}}(X)$ there is $c\in C$ such that $a\subseteq c$. >[!note] Remark >If we say that some set of countable sets, $S$, is stationary without mentioning any ambient set, we mean that it stationary in $\mathcal{P}_{\omega_{1}}(\cup S)$. The following connects the basic and generalized notions of stationarity. >[!note] Notation >For $A\subseteq\kappa$ and $B\subseteq\mathcal{P}_{\omega_{1}}(\kappa)$ denote >$\begin{aligned} \hat{A} & =\left\{ X\in\mathcal{P}_{\omega_{1}}(\kappa)\mid\sup X\cap\kappa\in A\right\} \\ \breve{B} & =\left\{ \sup X\cap\kappa\mid X\in B\right\} . >\end{aligned}$ >Notice that $\breve{\hat{A}}=A$, and $\hat{\breve{B}}\supseteq B$, but in the second case there might not be equality, as there may be more subsets having the same suprema as $B$. **Lemma.** Let $\kappa>\omega$ be regular, $A\subseteq E_{\omega}^{\kappa}$, $B\subseteq\mathcal{P}_{\omega_{1}}(\kappa)$. 1. $B$ is club in $\mathcal{P}_{\omega_{1}}(\kappa)$ $\implies$ $\breve{B}$ is $\sigma$-closed and unbounded in $\kappa$. 2. $A$ is $\sigma$-closed and unbounded in $\kappa$ $\implies$\hat{A}$ contains a club in $\mathcal{P}_{\omega_{1}}(\kappa)$. 3. $\hat{A}$ is club in $\mathcal{P}_{\omega_{1}}(\kappa)$ $\implies$ $A$ is $\sigma$-closed and unbounded in $\kappa$. 4. $B$ is stationary in $\mathcal{P}_{\omega_{1}}(\kappa)$ $\implies$ $\breve{B}$ is stationary in $\kappa$. 5. $A$ is stationary in $\kappa$ $\iff$ $\hat{A}$ is stationary in $\mathcal{P}_{\omega_{1}}(\kappa)$. An important property of stationary sets in $\mathcal{P}_{\omega_{1}}(\kappa)$ is that they project upwards and downwards, i.e. they form a *tower*: **Lemma.** Let $Y\supseteq X\ne\varnothing$. 1. If $S\subseteq\mathcal{P}_{\omega_{1}}\left(Y\right)$ is stationary then $S\mathord{\downarrow}X:=\left\{ Z\cap X\mid Z\in S\right\}$ is stationary. 2. If $S\subseteq\mathcal{P}_{\omega_{1}}\left(X\right)$ is stationary then $S\mathord{\uparrow}Y:=\left\{ Z\in\mathcal{P}_{\omega_{1}}\left(Y\right)\mid Z\cap X\in S\right\}$ is stationary.