Editing Club set (section)

==The closed unbounded filter==
{{main|Club filter}}
Let <math>\kappa \,</math> be a limit ordinal of uncountable [[cofinality]] <math>\lambda \,.</math> For some <math>\alpha < \lambda \,</math>, let <math>\langle C_\xi : \xi < \alpha\rangle \,</math> be a sequence of closed unbounded subsets of <math>\kappa \,.</math> Then <math>\bigcap_{\xi < \alpha} C_\xi \,</math> is also closed unbounded. To see this, one can note that an intersection of closed sets is always closed, so we just need to show that this intersection is unbounded. So fix any <math>\beta_0 < \kappa \,,</math> and for each ''n'' < &omega; choose from each <math>C_\xi \,</math> an element <math>\beta_{n+1}^\xi > \beta_{n} \,,</math> which is possible because each is unbounded. Since this is a collection of fewer than <math>\lambda \,</math> ordinals, all less than <math>\kappa \,,</math> their least upper bound must also be less than <math>\kappa \,,</math> so we can call it <math>\beta_{n+1} \,.</math> This process generates a countable sequence <math>\beta_0,\beta_1,\beta_2, \ldots \,.</math> The limit of this sequence must in fact also be the limit of the sequence <math>\beta_0^\xi,\beta_1^\xi,\beta_2^\xi, \ldots \,,</math> and since each <math>C_\xi \,</math> is closed and <math>\lambda \,</math> is uncountable, this limit must be in each <math>C_\xi \,,</math> and therefore this limit is an element of the intersection that is above <math>\beta_0 \,,</math> which shows that the intersection is unbounded. QED.

From this, it can be seen that if <math>\kappa \,</math> is a [[regular cardinal]], then <math>\{S \subseteq \kappa : \exists C \subseteq S \text{ such that } C \text{ is closed unbounded in } \kappa\}</math> is a non-principal <math>\kappa \,</math>-complete proper [[Filter (set theory)|filter]] on the set <math>\kappa</math> (that is, on the [[poset]] <math>(\wp(\kappa), \subseteq)</math>).

If <math>\kappa \,</math> is a regular cardinal then club sets are also closed under [[diagonal intersection]].

In fact, if <math>\kappa \,</math> is regular and <math>\mathcal{F} \,</math> is any filter on <math>\kappa \,,</math> closed under diagonal intersection, containing all sets of the form <math>\{\xi < \kappa : \xi \geq \alpha\} \,</math> for <math>\alpha < \kappa \,,</math> then <math>\mathcal{F} \,</math> must include all club sets.