club (original) (raw)

If κ is a cardinal then a set C⊆κ is closed iff for any S⊆C and α<κ, sup⁡(S∩α)=α then α∈C. (That is, if the limit of some sequence in C is less than κ then the limit is also in C.)

If κ is a cardinal and C⊆κ then C is unbounded if, for any α<κ, there is some β∈C such that α<β.

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