Editing Club set (section)

==Formal definition==

Formally, if <math>\kappa</math> is a limit ordinal, then a set <math>C\subseteq\kappa</math> is ''closed'' in <math>\kappa</math> [[if and only if]] for every <math>\alpha < \kappa,</math> if <math>\sup(C \cap \alpha) = \alpha \neq 0,</math> then <math>\alpha \in C.</math>  Thus, if the [[Limit of a sequence|limit of some sequence]] from <math>C</math> is less than <math>\kappa,</math> then the limit is also in <math>C.</math>

If <math>\kappa</math> is a limit ordinal and <math>C \subseteq \kappa</math> then <math>C</math> is '''unbounded''' in <math>\kappa</math> if for any <math>\alpha < \kappa,</math> there is some <math>\beta \in C</math> such that <math>\alpha < \beta.</math>

If a set is both closed and unbounded, then it is a '''club set'''. Closed [[proper class]]es 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.
If <math>\kappa</math> is an uncountable [[initial ordinal]], then the set of all limit ordinals <math>\alpha < \kappa</math> is closed unbounded in <math>\kappa.</math>  In fact a  club set is nothing else but the range of a [[normal function]]  (i.e. increasing and continuous).

More generally, if <math>X</math> is a nonempty set and <math>\lambda</math> is a [[Cardinal number|cardinal]], then <math>C \subseteq [X]^\lambda</math> (the set of subsets of <math>X</math> of cardinality <math>\lambda</math>) is ''club'' if every union of a subset of <math>C</math> is in <math>C</math> and every subset of <math>X</math> of cardinality less than <math>\lambda</math> is contained in some element of <math>C</math> (see [[stationary set]]).