= Club set
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= Closed unbounded set
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= Closed unbounded subset
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= Closed unbounded
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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>.
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