In mathematics, particularly in set theory, if is a regular uncountable cardinal then , the filter of all sets containing a club subset of , is a -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 containing is also in , since , and therefore anything containing it, contains a club set.
It is a -complete filter because the intersection of fewer than club sets is a club set. To see this, suppose is a sequence of club sets where . Obviously is closed, since any sequence which appears in appears in every , and therefore its limit is also in every . To show that it is unbounded, take some . Let be an increasing sequence with and for every . Such a sequence can be constructed, since every is unbounded. Since and is regular, the limit of this sequence is less than . We call it , 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 , is an increasing sequence contained in , and all these sequences have the same limit (the limit of ). This limit is then contained in every , and therefore , and is greater than .
To see that is closed under diagonal intersection, let , be a sequence of club sets, and let . To show is closed, suppose and . Then for each , for all . Since each is closed, for all , so . To show is unbounded, let , and define a sequence , as follows: , and is the minimal element of such that . Such an element exists since by the above, the intersection of club sets is club. Then and , since it is in each with .
- Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.