Club filter

In mathematics, particularly in set theory, if <math>\kappa<math> is a regular uncountable cardinal then <math>\operatorname{club}(\kappa)<math>, the filter of all sets containing a club subset of <math>\kappa<math>, is a <math>\kappa<math>-complete filter closed under diagonal intersection called the club filter.

To see that this is a filter, note that <math>\kappa\in\operatorname{club}(\kappa)<math> since it is thus both closed and unbounded (see club set). If <math>x\in\operatorname{club}(\kappa)<math> then any subset of <math>\kappa<math> containing <math>x<math> is also in <math>\operatorname{club}(\kappa)<math>, since <math>x<math>, and therefore anything containing it, contains a club set.

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

To see that <math>\operatorname{club}(\kappa)<math> is closed under diagonal intersection, let <math>\langle C_i\rangle<math>, <math>i<\kappa<math> be a sequence, and let <math>C=\Delta_{i<\kappa} C_i<math>. Since the diagonal intersection contains the intersection, obviously <math>C<math> is unbounded. Then suppose <math>S\subseteq C<math> and <math>\sup(S\cap\alpha)=\alpha<math>. Then <math>S\subseteq C_\beta<math> for every <math>\beta\geq\alpha<math>, and since each <math>C_\beta<math> is closed, <math>\alpha\in C_\beta<math>, so <math>\alpha\in C<math>.

This article incorporates material from club filter on PlanetMath, which is licensed under the GFDL.