{"history":[],"article":{"id":"675ba380-0a77-4fb2-8e9f-0b1b6ace2d43","title":"Club filter","slug":"club-filter","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), particularly in [set theory](https://bliptext.com/articles/set-theory), if \\kappa is a [regular](https://bliptext.com/articles/regular-[cardinal](https://bliptext.com/articles/cardinal-number)) [uncountable](https://bliptext.com/articles/uncountable) [cardinal](https://bliptext.com/articles/cardinal-number) then the [filter](https://bliptext.com/articles/filter-set-theory) of all [sets](https://bliptext.com/articles/set-mathematics) containing a [club subset](https://bliptext.com/articles/club-set) of \\kappa, is a \\kappa-complete [filter](https://bliptext.com/articles/filter-set-theory) closed under [diagonal intersection](https://bliptext.com/articles/diagonal-intersection) called the **club [filter](https://bliptext.com/articles/filter-set-theory)**.\nTo see that this is a filter, note that since it is thus both closed and unbounded (see [club set](https://bliptext.com/articles/club-set)).\nIf then any [subset](https://bliptext.com/articles/subset) of \\kappa containing x is also in since x, and therefore anything containing it, contains a club set.\nIt is a \\kappa-complete filter because the [intersection](https://bliptext.com/articles/intersection-set-theory) of fewer than \\kappa club sets is a club set.\nTo see this, suppose is a [sequence](https://bliptext.com/articles/sequence) of club sets where Obviously is closed, since any [sequence](https://bliptext.com/articles/sequence) which appears in C appears in every C_i, and therefore its [limit](https://bliptext.com/articles/direct-limit) is also in every C_i.\nTo show that it is unbounded, take some Let be an increasing sequence with and for every i < \\alpha.\nSuch a sequence can be constructed, since every C_i is unbounded.\nSince and \\kappa is regular, the limit of this sequence is less than \\kappa.\nWe call it \\beta_2, and define a new sequence similar to the previous sequence.\nWe can repeat this process, getting a sequence of sequences where each element of a sequence is greater than every member of the previous sequences.\nThen for each i < \\alpha, is an increasing sequence contained in C_i, and all these sequences have the same limit (the limit of ).\nThis limit is then contained in every C_i, and therefore C, and is greater than \\beta.\nTo see that is closed under diagonal intersection, let i < \\kappa be a sequence of club sets, and let To show C is closed, suppose and Then for each for all Since each C_\\beta is closed, for all so To show C is unbounded, let and define a sequence \\xi_i, i < \\omega as follows: and \\xi_{i+1} is the minimal element of such that Such an element exists since by the above, the intersection of \\xi_i club sets is club.\nThen and \\xi \\in C, since it is in each C_i with i < \\xi.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:32:50.453Z","updated_at":"2025-01-25T21:32:50.453Z","search_vector":"'club':1 'filter':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), particularly in [set theory](https://bliptext.com/articles/set-theory), if \\kappa is a [regular](https://bliptext.com/articles/regular-[cardinal](https://bliptext.com/articles/cardinal-number)) [uncountable](https://bliptext.com/articles/uncountable) [cardinal](https://bliptext.com/articles/cardinal-number) then the [filter](https://bliptext.com/articles/filter-set-theory) of all [sets](https://bliptext.com/articles/set-mathematics) containing a [club subset](https://bliptext.com/articles/club-set) of \\kappa, is a \\kappa-complete [filter](https://bliptext.com/articles/filter-set-theory) closed under [diagonal intersection](https://bliptext.com/articles/diagonal-intersection) called the **club [filter](https://bliptext.com/articles/filter-set-theory)**.\nTo see that this is a filter, note that since it is thus both closed and unbounded (see [club set](https://bliptext.com/articles/club-set)).\nIf then any [subset](https://bliptext.com/articles/subset) of \\kappa containing x is also in since x, and therefore anything containing it, contains a club set.\nIt is a \\kappa-complete filter because the [intersection](https://bliptext.com/articles/intersection-set-theory) of fewer than \\kappa club sets is a club set.\nTo see this, suppose is a [sequence](https://bliptext.com/articles/sequence) of club sets where Obviously is closed, since any [sequence](https://bliptext.com/articles/sequence) which appears in C appears in every C_i, and therefore its [limit](https://bliptext.com/articles/direct-limit) is also in every C_i.\nTo show that it is unbounded, take some Let be an increasing sequence with and for every i < \\alpha.\nSuch a sequence can be constructed, since every C_i is unbounded.\nSince and \\kappa is regular, the limit of this sequence is less than \\kappa.\nWe call it \\beta_2, and define a new sequence similar to the previous sequence.\nWe can repeat this process, getting a sequence of sequences where each element of a sequence is greater than every member of the previous sequences.\nThen for each i < \\alpha, is an increasing sequence contained in C_i, and all these sequences have the same limit (the limit of ).\nThis limit is then contained in every C_i, and therefore C, and is greater than \\beta.\nTo see that is closed under diagonal intersection, let i < \\kappa be a sequence of club sets, and let To show C is closed, suppose and Then for each for all Since each C_\\beta is closed, for all so To show C is unbounded, let and define a sequence \\xi_i, i < \\omega as follows: and \\xi_{i+1} is the minimal element of such that Such an element exists since by the above, the intersection of \\xi_i club sets is club.\nThen and \\xi \\in C, since it is in each C_i with i < \\xi.","revisionCount":0},"hasMore":false,"nextCursor":null}

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