# Is there an abstract theory of club sets and stationary sets?

In set theory, there are several distinct notions of club sets, stationary sets, diagonal intersection, regressive function, normal filters, normal ultrafilters, etc. I am wondering if there is an abstract general theory of stationary that encapsulates all or most of these notions at the same time. Consider the following examples.

Example 0: Suppose that $$X$$ is a topological space and $$\infty\in X$$ is a non-isolated point. Then we say that a subset $$C\subseteq X$$ is unbounded if $$C\cap U\setminus\{\infty\}\neq\emptyset$$ is non-empty for each neighborhood $$U$$ of $$\infty$$. A club set is defined to be a closed unbounded set. For most spaces $$X$$ along with $$\infty\in X$$, this notion of clubness is unsatisfactory since the intersection of two club sets is typically no longer a club set, but there are spaces where the intersection of club sets is still a club set.

Example 1: Suppose that $$\kappa$$ is an uncountable cardinal. Then a subset $$C\subseteq\kappa$$ is a club set if $$C$$ is closed when $$\kappa$$ is given the order topology and where $$C\cap(\kappa\setminus\alpha)\neq\emptyset$$ for each $$\alpha<\kappa$$.

Example 2: Suppose that $$\kappa$$ is an uncountable regular cardinal and $$\lambda$$ is a cardinal with $$\lambda\geq\kappa$$. Let $$P_{\kappa}(\lambda)$$ be the collection of all subsets of $$R\subseteq\lambda$$ with $$|R|<\kappa$$. A subset $$C\subseteq P_{\kappa}(\lambda)$$ is said to be closed if whenever $$\gamma<\lambda$$ and $$R_{\alpha}\in C$$ for $$\alpha<\gamma$$ and $$R_{\alpha}\subseteq R_{\beta}$$ whenever $$\alpha<\beta<\gamma$$ we have $$\bigcup_{\alpha<\gamma}R_{\alpha}$$. A subset $$C\subseteq P_{\kappa}(\lambda)$$ is unbounded if for each $$R\in P_{\kappa}(\lambda)$$, there is some $$S\in C$$ with $$R\subseteq S$$. A club set is again defined to be a closed unbounded set.

Example 3: Suppose again that $$\kappa$$ is an inaccessible cardinal and $$\lambda$$ is a cardinal with $$\lambda\geq\kappa$$. If $$f:P_{\kappa}(\lambda)\rightarrow P_{\kappa}(\lambda)$$, then define $$C_{f}\subseteq P_{\kappa}(\lambda)$$ by letting $$C_{f}=\{x\in P_{\kappa}(\lambda):x\cap\kappa\neq\emptyset, f[P_{|x\cap\kappa|}(x)]\subseteq P_{|x\cap\kappa|}(x)\}.$$ We say that a subset $$C\subseteq P_{\kappa}(\lambda)$$ is weakly closed if $$C=C_{f}$$ for some $$f$$.

In each of these examples (except Example 0), we have notions of stationary sets, normal filters, etc. Is there a general abstract theory of that encapsulates all of examples of notions of club and stationary sets? If there is no general abstract theory, then are there any reasons why no abstract theory has been formulated? It seems like such an abstract theory may be formulated in terms of ordered sets and general topology.

• About 45 minutes after posting this question, I just recalled this paper that attempts to classify large cardinal axioms using filter spaces and these filter spaces give an abstract notion of stationarity. Has such an abstract theory of filter spaces been developed further by anyone? sciencedirect.com/science/article/pii/0168007289900092 Mar 10 '19 at 19:37
• In the above paper, the authors ask the question as to what the correct notion of a club set will be in a filter space, so it seems like the notion of a filter space may need to be modified to encapsulate club sets. Mar 10 '19 at 20:21
• You may also look at the answer given at Stationarity and Fodor's lemma for a (nice) poset? Mar 11 '19 at 3:56
• The following notion is due to Woodin. Let $Z$ be any set and let $X = \bigcup Z$. We say $Z$ is stationary if for every function $F : X^{<\omega} \to X$, there is $z \in Z$ such that $z$ is closed under $F$. It is a theorem of Kueker that the notion of club for $P_\kappa(\lambda)$ mentioned in Example 2 is equivalent to: "$C$ is in the club filter iff there is a function $F : \lambda^{<\omega} \to \lambda$ such that $C$ contains the set of $z$ such that $z \cap \kappa \in \kappa$ and $z$ is closed under $F$." Mar 11 '19 at 12:59
• It is well-known that the notion of club in Example 2 is equivalent to that in Example 1, when restricted to the club $\kappa \subseteq P_\kappa(\kappa)$. I don't know about example 3. Is it non-equivalent to the usual notion? Mar 11 '19 at 13:02

## 1 Answer

Stationary logic goes very far in the abstract analysis of stationarity; this article by Barwise, Makkai, and Kaufmann (1978) started the subject.

See also this MO post and its answer.