In mathematics, particularly in set theory, if κ is a regular uncountable cardinal then club ( κ ) , the filter of all sets containing a club subset of κ , is a κ -complete filter closed under diagonal intersection called the club filter.
To see that this is a filter, note that κ ∈ club ( κ ) since it is thus both closed and unbounded (see club set). If x ∈ club ( κ ) then any subset of κ containing x is also in club ( κ ) , since x , and therefore anything containing it, contains a club set.
It is a κ -complete filter because the intersection of fewer than κ club sets is a club set. To see this, suppose ⟨ C i ⟩ i < α is a sequence of club sets where α < κ . Obviously C = ⋂ C i is closed, since any sequence which appears in C appears in every C i , and therefore its limit is also in every C i . To show that it is unbounded, take some β < κ . Let ⟨ β 1 , i ⟩ be an increasing sequence with β 1 , 1 > β and β 1 , i ∈ C i for every i < α . Such a sequence can be constructed, since every C i is unbounded. Since α < κ and κ is regular, the limit of this sequence is less than κ . We call it β 2 , and define a new sequence ⟨ β 2 , i ⟩ similar to the previous sequence. We can repeat this process, getting a sequence of sequences ⟨ β j , i ⟩ where each element of a sequence is greater than every member of the previous sequences. Then for each i < α , ⟨ β j , i ⟩ is an increasing sequence contained in C i , and all these sequences have the same limit (the limit of ⟨ β j , i ⟩ ). This limit is then contained in every C i , and therefore C , and is greater than β .
To see that club ( κ ) is closed under diagonal intersection, let ⟨ C i ⟩ , i < κ be a sequence of club sets, and let C = Δ i < κ C i . To show C is closed, suppose S ⊆ α < κ and ⋃ S = α . Then for each γ ∈ S , γ ∈ C β for all β < γ . Since each C β is closed, α ∈ C β for all β < α , so α ∈ C . To show C is unbounded, let α < κ , and define a sequence ξ i , i < ω as follows: ξ 0 = α , and ξ i + 1 is the minimal element of ⋂ γ < ξ i C γ such that ξ i + 1 > ξ i . Such an element exists since by the above, the intersection of ξ i club sets is club. Then ξ = ⋃ i < ω ξ i > α and ξ ∈ C , since it is in each C i with i < ξ .
