- Source: 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 .}
See also
Clubsuit – in set theory, the combinatorial principle that, for every stationary 𝑆⊂ω₁, there exists a sequence of sets 𝐴_𝛿 (𝛿∈𝑆) such that 𝐴_𝛿 is a cofinal subset of 𝛿 and every unbounded subset of ω₁ is contained in some 𝐴_𝛿Pages displaying wikidata descriptions as a fallback
Filter (mathematics) – In mathematics, a special subset of a partially ordered set
Stationary set – Set-theoretic concept
References
This article incorporates material from club filter on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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