The club filter and nonstationary ideal

In ZFC, assume $\mathrm{cf}(\theta )>\omega$. The club filter is ${\mathcal{C}}_{\theta }=\{A\subseteq \theta :\mathrm{\exists }C\subseteq A(C\text{ club in }\theta )\}$. A set $S\subseteq \theta$ is stationary if it meets every club. It is nonstationary if disjoint from some club; these sets form ${\mathrm{N}\mathrm{S}}_{\theta }$.

A proper filter contains the ambient set, excludes the empty set, is upward closed, and is closed under finite intersections. These hold for ${\mathcal{C}}_{\theta }$: the ambient set is club, clubs are nonempty, and the small-intersection theorem gives a club inside each finite intersection. In fact it is closed under intersections of fewer than $\mathrm{cf}(\theta )$ members: in ZFC choose a witnessing club for each member, then intersect them.

An ideal contains the empty set, is downward closed and closed under finite unions. Here $A\in {\mathrm{N}\mathrm{S}}_{\theta }$ iff $\theta \setminus A\in {\mathcal{C}}_{\theta }$, so complements give these axioms and closure under unions of fewer than $\mathrm{cf}(\theta )$ members. The filter contains all supersets of clubs, which need not themselves be closed.