Definitions

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.

Formally, if $kappa$ is a limit ordinal, then a set $Csubseteqkappa$ is closed in $kappa$ if and only if for every

If $kappa$ is a limit ordinal and $Csubseteqkappa$ then $C$ is unbounded in $kappa$ if and only if for any

If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).

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.