Club set

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 (see below) relative to the limit ordinal. The name club is a contraction of "closed and unbounded".

Formally, if ${\displaystyle \kappa }$ is a limit ordinal, then a set ${\displaystyle C\subseteq \kappa }$ is closed in ${\displaystyle \kappa }$ if and only if for every ${\displaystyle \alpha <\kappa }$, if ${\displaystyle \sup(C\cap \alpha )=\alpha \neq 0}$, then ${\displaystyle \alpha \in C}$. Thus, if the limit of some sequence from ${\displaystyle C}$ is less than ${\displaystyle \kappa }$, then the limit is also in ${\displaystyle C}$.

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