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In mathematics, particularly in
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, if \kappa is a
regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal numb ...
cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **'' Cardinalis'', genus of cardinal in the family Cardinalidae **'' Cardinalis cardinalis'', or northern cardinal, ...
then \operatorname(\kappa), the filter of all sets containing a club subset of \kappa, is a \kappa-complete filter closed under
diagonal intersection Diagonal intersection is a term used in mathematics, especially in set theory. If \displaystyle\delta is an ordinal number and \displaystyle\langle X_\alpha \mid \alpha<\delta\rangle is a
called the club filter. To see that this is a filter, note that \kappa \in \operatorname(\kappa) since it is thus both closed and unbounded (see
club set In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name ''club'' is a contraction ...
). If x\in\operatorname(\kappa) then any
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of \kappa containing x is also in \operatorname(\kappa), since x, and therefore anything containing it, contains a club set. It is a \kappa-complete filter because the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of fewer than \kappa club sets is a club set. To see this, suppose \langle C_i\rangle_ is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of club sets where \alpha < \kappa. Obviously C = \bigcap C_i is closed, since any sequence which appears in C appears in every C_i, and therefore its
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
is also in every C_i. To show that it is unbounded, take some \beta < \kappa. Let \langle \beta_\rangle be an increasing sequence with \beta_ > \beta and \beta_ \in C_i for every i < \alpha. Such a sequence can be constructed, since every C_i is unbounded. Since \alpha < \kappa and \kappa is regular, the limit of this sequence is less than \kappa. We call it \beta_2, and define a new sequence \langle\beta_\rangle similar to the previous sequence. We can repeat this process, getting a sequence of sequences \langle\beta_\rangle where each element of a sequence is greater than every member of the previous sequences. Then for each i < \alpha, \langle\beta_\rangle is an increasing sequence contained in C_i, and all these sequences have the same limit (the limit of \langle\beta_\rangle). This limit is then contained in every C_i, and therefore C, and is greater than \beta. To see that \operatorname(\kappa) is closed under diagonal intersection, let \langle C_i\rangle, i < \kappa be a sequence of club sets, and let C = \Delta_ C_i. To show C is closed, suppose S\subseteq \alpha < \kappa and \bigcup S = \alpha. Then for each \gamma \in S, \gamma \in C_\beta for all \beta < \gamma. Since each C_\beta is closed, \alpha \in C_\beta for all \beta < \alpha, so \alpha \in C. To show C is unbounded, let \alpha < \kappa, and define a sequence \xi_i, i < \omega as follows: \xi_0 = \alpha, and \xi_ is the minimal element of \bigcap_ C_\gamma such that \xi_ > \xi_i. Such an element exists since by the above, the intersection of \xi_i club sets is club. Then \xi = \bigcup_ \xi_i > \alpha and \xi \in C, since it is in each C_i with i < \xi.


See also

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References

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