Editing Club set

{{Short description|Set theory concept}}
In [[mathematics]], particularly in [[mathematical logic]] and [[set theory]], a '''club set''' is a subset of a [[limit ordinal]] that is [[closed set|closed]] under the [[order topology]], and is unbounded (see below) relative to the limit ordinal.  The name ''club'' is a contraction of "closed and unbounded".

==Formal definition==

Formally, if <math>\kappa</math> is a limit ordinal, then a set <math>C\subseteq\kappa</math> is ''closed'' in <math>\kappa</math> [[if and only if]] for every <math>\alpha < \kappa,</math> if <math>\sup(C \cap \alpha) = \alpha \neq 0,</math> then <math>\alpha \in C.</math>  Thus, if the [[Limit of a sequence|limit of some sequence]] from <math>C</math> is less than <math>\kappa,</math> then the limit is also in <math>C.</math>

If <math>\kappa</math> is a limit ordinal and <math>C \subseteq \kappa</math> then <math>C</math> is '''unbounded''' in <math>\kappa</math> if for any <math>\alpha < \kappa,</math> there is some <math>\beta \in C</math> such that <math>\alpha < \beta.</math>

If a set is both closed and unbounded, then it is a '''club set'''. Closed [[proper class]]es are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).

For example, the set of all [[countable]] limit ordinals is a club set with respect to the [[first uncountable ordinal]]; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded.
If <math>\kappa</math> is an uncountable [[initial ordinal]], then the set of all limit ordinals <math>\alpha < \kappa</math> is closed unbounded in <math>\kappa.</math>  In fact a  club set is nothing else but the range of a [[normal function]]  (i.e. increasing and continuous).

More generally, if <math>X</math> is a nonempty set and <math>\lambda</math> is a [[Cardinal number|cardinal]], then <math>C \subseteq [X]^\lambda</math> (the set of subsets of <math>X</math> of cardinality <math>\lambda</math>) is ''club'' if every union of a subset of <math>C</math> is in <math>C</math> and every subset of <math>X</math> of cardinality less than <math>\lambda</math> is contained in some element of <math>C</math> (see [[stationary set]]).

==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.

==See also==

* {{annotated link|Clubsuit}}
* {{annotated link|Filter (mathematics)}}
* {{annotated link|Filters in topology}}
* {{annotated link|Stationary set}}

==References==
{{reflist}}
{{refbegin}}
* [[Thomas Jech|Jech, Thomas]], 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''.  Springer.  {{ISBN|3-540-44085-2}}.
* [[Azriel Levy|Lévy, Azriel]] (1979) ''Basic Set Theory'', Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover. {{ISBN|0-486-42079-5}}
* {{PlanetMath attribution|id=3227|title=Club}}
{{refend}}

{{Order theory}}

[[Category:Ordinal numbers]]
[[Category:Set theory]]