Club filter completeness
= Club filter completeness
{title2=$\mathcal D_\kappa\text{ is }\kappa\text{-complete}$}
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.