Club set

For a limit ordinal $\kappa$, a subset $C\subseteq \kappa$ is club if it is unbounded and contains each limit below $\kappa$ of an increasing sequence from $C$. At an uncountable regular cardinal $\kappa$, intersections of fewer than $\kappa$ club sets are club: above a starting point cycle through the sets and take suprema, using regularity to stay below $\kappa$; closure puts the resulting limit in all of them. Club sets provide the reflecting ranks in worldly cardinals below an inaccessible cardinal.

An unbounded class in set theory of ordinals is closed if every nonempty set-sized increasing sequence of its members has its supremum in the class. This is the proper-class analogue of a club set. The reflection theorem for definable hierarchies gives such a class for every finite collection of first-order formulas.