In mathematics, particularly in set theory, if κ is a regular uncountable cardinal then , 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 since it is thus both closed and unbounded (see club set). If then any subset of κ containing x is also in , 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 is a sequence of club sets where α < κ. Obviously 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 be an increasing sequence with β_1,1 > β and 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 β₂, and define a new sequence similar to the previous sequence. We can repeat this process, getting a sequence of sequences where each element of a sequence is greater than every member of the previous sequences. Then for each i < α, is an increasing sequence contained in C_i, and all these sequences have the same limit (the limit of ). This limit is then contained in every C_i, and therefore C, and is greater than β.

To see that is closed under diagonal intersection, let , i < κ be a sequence, and let C = Δ_{i < κ}C_i. Since the diagonal intersection contains the intersection, obviously C is unbounded. Then suppose and . Then for every , and since each C_β is closed, , so .