Club set

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In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded.

Formally, if [\kappa] is a limit ordinal, then a set [C\subseteq\kappa] is closed in [\kappa] if and only if for every [\alpha<\kappa], if [\sup(C\cap \alpha)=\alpha\ne0], then [\alpha\in C]. Thus, if the limit of some sequence in [C] is less than [\kappa], then the limit is also in [C].

If [\kappa] is a limit ordinal and [C\subseteq\kappa] then [C] is unbounded in [\kappa] if and only if for any [\alpha<\kappa], there is some [\beta\in C] such that [\alpha<\beta].

If a set is both closed and unbounded, then it is a club set.

For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor bounded.