Club filter
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

See also

References

 Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. . Set theory