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| In [[mathematics]], particularly in [[set theory]], if <math>\kappa</math> is a [[Regular cardinal|regular]] [[uncountable]] [[Cardinal number|cardinal]] then <math>\operatorname{club}(\kappa)</math>, the [[Filter (mathematics)|filter]] of all [[Set (mathematics)|sets]] containing a [[Club set|club subset]] of <math>\kappa</math>, is a <math>\kappa</math>-complete filter closed under [[diagonal intersection]] called the '''club filter'''.
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| 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.
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| It is a <math>\kappa</math>-complete filter because the [[Intersection (set theory)|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 [[Direct limit|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>.
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| 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 of club sets, and let <math>C=\Delta_{i<\kappa} C_i</math>. To show <math>C</math> is closed, suppose <math>S\subseteq \alpha<\kappa</math> and <math>\bigcup S=\alpha</math>. Then for each <math>\gamma\in S</math>, <math>\gamma\in C_\beta</math> for all <math>\beta<\gamma</math>. Since each <math>C_\beta</math> is closed, <math>\alpha\in C_\beta</math> for all <math>\beta<\alpha</math>, so <math>\alpha\in C</math>. To show <math>C</math> is unbounded, let <math>\alpha<\kappa</math>, and define a sequence <math>\xi_i</math>, <math>i<\omega</math> as follows: <math>\xi_0=\alpha</math>, and <math>\xi_{i+1}</math> is the minimal element of <math>\bigcap_{\gamma<\xi_i}C_\gamma</math> such that <math>\xi_{i+1}>\xi_i</math>. Such an element exists since by the above, the intersection of <math>\xi_i</math> club sets is club. Then <math>\xi=\bigcup_{i<\omega}\xi_i>\alpha</math> and <math>\xi\in C</math>, since it is in each <math>C_i</math> with <math>i<\xi</math>.
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| ==References==
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| *Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
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| {{PlanetMath attribution|id=3231|title=club filter}}
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| [[Category:Set theory]]
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