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

If $κ$ is a limit ordinal and $C\subseteq\kappa$ then $C$ is unbounded in $κ$ if and only if for any $α < κ$ , there is some $\beta\in C$ such that $α < β$ .

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 unbounded.