Club filter

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

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

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

To see that club ⁡ ( κ ) {\displaystyle \operatorname {club} (\kappa )} is closed under diagonal intersection, let ⟨ C i ⟩ , {\displaystyle \langle C_{i}\rangle ,} i < κ {\displaystyle i<\kappa } be a sequence of club sets, and let C = Δ i < κ C i . {\displaystyle C=\Delta _{i<\kappa }C_{i}.} To show C {\displaystyle C} is closed, suppose S ⊆ α < κ {\displaystyle S\subseteq \alpha <\kappa } and ⋃ S = α . {\displaystyle \bigcup S=\alpha .} Then for each γ ∈ S , {\displaystyle \gamma \in S,} γ ∈ C β {\displaystyle \gamma \in C_{\beta }} for all β < γ . {\displaystyle \beta <\gamma .} Since each C β {\displaystyle C_{\beta }} is closed, α ∈ C β {\displaystyle \alpha \in C_{\beta }} for all β < α , {\displaystyle \beta <\alpha ,} so α ∈ C . {\displaystyle \alpha \in C.} To show C {\displaystyle C} is unbounded, let α < κ , {\displaystyle \alpha <\kappa ,} and define a sequence ξ i , {\displaystyle \xi _{i},} i < ω {\displaystyle i<\omega } as follows: ξ 0 = α , {\displaystyle \xi _{0}=\alpha ,} and ξ i + 1 {\displaystyle \xi _{i+1}} is the minimal element of ⋂ γ < ξ i C γ {\displaystyle \bigcap _{\gamma <\xi _{i}}C_{\gamma }} such that ξ i + 1 > ξ i . {\displaystyle \xi _{i+1}>\xi _{i}.} Such an element exists since by the above, the intersection of ξ i {\displaystyle \xi _{i}} club sets is club. Then ξ = ⋃ i < ω ξ i > α {\displaystyle \xi =\bigcup _{i<\omega }\xi _{i}>\alpha } and ξ ∈ C , {\displaystyle \xi \in C,} since it is in each C i {\displaystyle C_{i}} with i < ξ . {\displaystyle i<\xi .}

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

This article incorporates material from club filter on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.