Club filter

In mathematics, particularly in set theory, if [math]\displaystyle{ \kappa }[/math] is a regular uncountable cardinal then [math]\displaystyle{ \operatorname{club}(\kappa), }[/math] the filter of all sets containing a club subset of [math]\displaystyle{ \kappa, }[/math] is a [math]\displaystyle{ \kappa }[/math]-complete filter closed under diagonal intersection called the club filter. To see that this is a filter, note that [math]\displaystyle{ \kappa \in \operatorname{club}(\kappa) }[/math] since it is thus both closed and unbounded (see club set). If [math]\displaystyle{ x\in\operatorname{club}(\kappa) }[/math] then any subset of [math]\displaystyle{ \kappa }[/math] containing [math]\displaystyle{ x }[/math] is also in [math]\displaystyle{ \operatorname{club}(\kappa), }[/math] since [math]\displaystyle{ x, }[/math] and therefore anything containing it, contains a club set.

It is a [math]\displaystyle{ \kappa }[/math]-complete filter because the intersection of fewer than [math]\displaystyle{ \kappa }[/math] club sets is a club set. To see this, suppose [math]\displaystyle{ \langle C_i\rangle_{i\lt \alpha} }[/math] is a sequence of club sets where [math]\displaystyle{ \alpha \lt \kappa. }[/math] Obviously [math]\displaystyle{ C = \bigcap C_i }[/math] is closed, since any sequence which appears in [math]\displaystyle{ C }[/math] appears in every [math]\displaystyle{ C_i, }[/math] and therefore its limit is also in every [math]\displaystyle{ C_i. }[/math] To show that it is unbounded, take some [math]\displaystyle{ \beta \lt \kappa. }[/math] Let [math]\displaystyle{ \langle \beta_{1,i}\rangle }[/math] be an increasing sequence with [math]\displaystyle{ \beta_{1,1} \gt \beta }[/math] and [math]\displaystyle{ \beta_{1,i} \in C_i }[/math] for every [math]\displaystyle{ i \lt \alpha. }[/math] Such a sequence can be constructed, since every [math]\displaystyle{ C_i }[/math] is unbounded. Since [math]\displaystyle{ \alpha \lt \kappa }[/math] and [math]\displaystyle{ \kappa }[/math] is regular, the limit of this sequence is less than [math]\displaystyle{ \kappa. }[/math] We call it [math]\displaystyle{ \beta_2, }[/math] and define a new sequence [math]\displaystyle{ \langle\beta_{2,i}\rangle }[/math] similar to the previous sequence. We can repeat this process, getting a sequence of sequences [math]\displaystyle{ \langle\beta_{j,i}\rangle }[/math] where each element of a sequence is greater than every member of the previous sequences. Then for each [math]\displaystyle{ i \lt \alpha, }[/math] [math]\displaystyle{ \langle\beta_{j,i}\rangle }[/math] is an increasing sequence contained in [math]\displaystyle{ C_i, }[/math] and all these sequences have the same limit (the limit of [math]\displaystyle{ \langle\beta_{j,i}\rangle }[/math]). This limit is then contained in every [math]\displaystyle{ C_i, }[/math] and therefore [math]\displaystyle{ C, }[/math] and is greater than [math]\displaystyle{ \beta. }[/math]

To see that [math]\displaystyle{ \operatorname{club}(\kappa) }[/math] is closed under diagonal intersection, let [math]\displaystyle{ \langle C_i\rangle, }[/math] [math]\displaystyle{ i \lt \kappa }[/math] be a sequence of club sets, and let [math]\displaystyle{ C = \Delta_{i\lt \kappa} C_i. }[/math] To show [math]\displaystyle{ C }[/math] is closed, suppose [math]\displaystyle{ S\subseteq \alpha \lt \kappa }[/math] and [math]\displaystyle{ \bigcup S = \alpha. }[/math] Then for each [math]\displaystyle{ \gamma \in S, }[/math] [math]\displaystyle{ \gamma \in C_\beta }[/math] for all [math]\displaystyle{ \beta \lt \gamma. }[/math] Since each [math]\displaystyle{ C_\beta }[/math] is closed, [math]\displaystyle{ \alpha \in C_\beta }[/math] for all [math]\displaystyle{ \beta \lt \alpha, }[/math] so [math]\displaystyle{ \alpha \in C. }[/math] To show [math]\displaystyle{ C }[/math] is unbounded, let [math]\displaystyle{ \alpha \lt \kappa, }[/math] and define a sequence [math]\displaystyle{ \xi_i, }[/math] [math]\displaystyle{ i \lt \omega }[/math] as follows: [math]\displaystyle{ \xi_0 = \alpha, }[/math] and [math]\displaystyle{ \xi_{i+1} }[/math] is the minimal element of [math]\displaystyle{ \bigcap_{\gamma\lt \xi_i} C_\gamma }[/math] such that [math]\displaystyle{ \xi_{i+1} \gt \xi_i. }[/math] Such an element exists since by the above, the intersection of [math]\displaystyle{ \xi_i }[/math] club sets is club. Then [math]\displaystyle{ \xi = \bigcup_{i\lt \omega} \xi_i \gt \alpha }[/math] and [math]\displaystyle{ \xi \in C, }[/math] since it is in each [math]\displaystyle{ C_i }[/math] with [math]\displaystyle{ i \lt \xi. }[/math]

See also

• Clubsuit
• Filter (mathematics) – In mathematics, a special subset of a partially ordered set
• Stationary set – Set-theoretic concept

References

• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.