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.



Is there an abstract theory of club sets and stationary sets?
*Stationary logic* goes very far in the abstract analysis of stationarity; [this article](https://core.ac.uk/download/pdf/81121517.pdf) by Barwise, Makkai, and Kaufmann (1978) started the subject.


See also [this MO post](https://math.stackexchange.com/questions/2371500/definition-generalized-logics-and-stationary-logics) and its answer.


