Club filter (${\mathcal{D}}_{\kappa }=\{A\subseteq \kappa :\exists \text{ club }C\subseteq A\}$)

On a regular uncountable cardinal number $\kappa$, the club filter consists of all subsets containing a club set. It is a kappa-complete filter by club filter completeness. Its members are stationary sets, although a stationary set need not be a filter member, and a filter member need not itself be closed.

For any regular uncountable cardinal number $\kappa$, the set $\kappa \setminus \{\omega \}$ contains the final-segment club set $[\omega +1,\kappa )$, so belongs to the club filter and is a stationary set. It is not closed, since its finite ordinals have supremum $\omega$, which is missing.

The club filter on a regular uncountable cardinal number $\kappa$ is $\kappa$-complete. Intersections of fewer than $\kappa$ club sets are closed. For unboundedness, repeatedly step past a point of each club and take a countable supremum; regularity keeps all suprema below $\kappa$, and closure puts the final supremum in every club.