In mathematics, particularly in set theory, if
κ
is a regular uncountable cardinal then
club
(
κ
)
, the filter of all sets containing a club subset of
κ
, is a
κ
-complete filter closed under diagonal intersection called the club filter.
To see that this is a filter, note that
κ
∈
club
(
κ
)
since it is thus both closed and unbounded (see club set). If
x
∈
club
(
κ
)
then any subset of
κ
containing
x
is also in
club
(
κ
)
, since
x
, and therefore anything containing it, contains a club set.
It is a
κ
-complete filter because the intersection of fewer than
κ
club sets is a club set. To see this, suppose
⟨
C
i
⟩
i
<
α
is a sequence of club sets where
α
<
κ
. Obviously
C
=
⋂
C
i
is closed, since any sequence which appears in
C
appears in every
C
i
, and therefore its limit is also in every
C
i
. To show that it is unbounded, take some
β
<
κ
. Let
⟨
β
1
,
i
⟩
be an increasing sequence with
β
1
,
1
>
β
and
β
1
,
i
∈
C
i
for every
i
<
α
. Such a sequence can be constructed, since every
C
i
is unbounded. Since
α
<
κ
and
κ
is regular, the limit of this sequence is less than
κ
. We call it
β
2
, and define a new sequence
⟨
β
2
,
i
⟩
similar to the previous sequence. We can repeat this process, getting a sequence of sequences
⟨
β
j
,
i
⟩
where each element of a sequence is greater than every member of the previous sequences. Then for each
i
<
α
,
⟨
β
j
,
i
⟩
is an increasing sequence contained in
C
i
, and all these sequences have the same limit (the limit of
⟨
β
j
,
i
⟩
). This limit is then contained in every
C
i
, and therefore
C
, and is greater than
β
.
To see that
club
(
κ
)
is closed under diagonal intersection, let
⟨
C
i
⟩
,
i
<
κ
be a sequence of club sets, and let
C
=
Δ
i
<
κ
C
i
. To show
C
is closed, suppose
S
⊆
α
<
κ
and
⋃
S
=
α
. Then for each
γ
∈
S
,
γ
∈
C
β
for all
β
<
γ
. Since each
C
β
is closed,
α
∈
C
β
for all
β
<
α
, so
α
∈
C
. To show
C
is unbounded, let
α
<
κ
, and define a sequence
ξ
i
,
i
<
ω
as follows:
ξ
0
=
α
, and
ξ
i
+
1
is the minimal element of
⋂
γ
<
ξ
i
C
γ
such that
ξ
i
+
1
>
ξ
i
. Such an element exists since by the above, the intersection of
ξ
i
club sets is club. Then
ξ
=
⋃
i
<
ω
ξ
i
>
α
and
ξ
∈
C
, since it is in each
C
i
with
i
<
ξ
.
