As club amount is in the set theory a subset of a limit ordinal referred to the complete and unlimited (engl. Cl OSED and u n b is ounded).
definition
Let be a Limesordinal number. A subset is called
λ
{\ displaystyle \ lambda}
x
⊆
λ
{\ displaystyle x \ subseteq \ lambda}
complete if for any sequence of the following applies:
⟨
α
ξ
∈
x
∣
ξ
<
μ
⟩
{\ displaystyle \ langle \ alpha _ {\ xi} \ in x \ mid \ xi <\ mu \ rangle}
x
{\ displaystyle x}
lim
ξ
→
μ
α
ξ
=
δ
∈
λ
⇒
δ
∈
x
,
{\ displaystyle \ lim _ {\ xi \ to \ mu} \ alpha _ {\ xi} = \ delta \ in \ lambda \ Rightarrow \ delta \ in x,}
unlimited if all one does with .
α
∈
λ
{\ displaystyle \ alpha \ in \ lambda}
β
∈
x
{\ displaystyle \ beta \ in x}
α
≤
β
{\ displaystyle \ alpha \ leq \ beta}
x
{\ displaystyle x}
is called the club set if it is both completed and unlimited.
x
{\ displaystyle x}
Examples
For the condition of seclusion is trivially fulfilled because there are no limit ordinal numbers below ; club sets of are only unbounded, i.e. H. infinite subsets of the natural numbers.
λ
=
ω
{\ displaystyle \ lambda = \ omega}
ω
{\ displaystyle \ omega}
ω
{\ displaystyle \ omega}
If one and the class of ordinal numbers are understood as topological spaces by means of the order topology , then the image of every continuous, monotonously increasing function is a club set.
λ
{\ displaystyle \ lambda}
Ord
{\ displaystyle \ operatorname {Ord}}
f
:
λ
→
Ord
{\ displaystyle f \ colon \ lambda \ to \ operatorname {Ord}}
The club filter
Is the cofinality the limit cardinal number uncountable, so the intersection of two club-back amount is an amount club. If you set , a filter forms , the club filter . It has the following properties, among others:
λ
{\ displaystyle \ lambda}
cf.
λ
>
ω
{\ displaystyle \ operatorname {cf} \ lambda> \ omega}
C.
λ
=
{
x
⊆
λ
∣
∃
C.
⊆
x
C.
club
}
{\ displaystyle {\ mathcal {C}} _ {\ lambda} = \ {x \ subseteq \ lambda \ mid \ exists C \ subseteq x \ C {\ text {club}} \}}
C.
λ
{\ displaystyle {\ mathcal {C _ {\ lambda}}}}
C.
λ
{\ displaystyle {\ mathcal {C}} _ {\ lambda}}
is -complete: Is and for , then applies
cf.
λ
{\ displaystyle \ operatorname {cf} \ lambda}
γ
∈
cf.
λ
{\ displaystyle \ gamma \ in \ operatorname {cf} \ lambda}
C.
α
∈
C.
λ
{\ displaystyle C _ {\ alpha} \ in {\ mathcal {C}} _ {\ lambda}}
α
∈
γ
{\ displaystyle \ alpha \ in \ gamma}
⋂
α
∈
γ
C.
α
∈
C.
λ
.
{\ displaystyle \ textstyle \ bigcap \ limits _ {\ alpha \ in \ gamma} C _ {\ alpha} \ in {\ mathcal {C}} _ {\ lambda}.}
If a regular cardinal number is , then so -called diagonal cuts are closed : If a family is made up of club sets , so is
λ
{\ displaystyle \ lambda}
C.
λ
{\ displaystyle {\ mathcal {C}} _ {\ lambda}}
⟨
C.
α
∣
α
∈
λ
⟩
{\ displaystyle \ langle C _ {\ alpha} \ mid \ alpha \ in \ lambda \ rangle}
C.
λ
{\ displaystyle {\ mathcal {C}} _ {\ lambda}}
△
α
∈
λ
C.
α
: =
{
β
∈
λ
∣
β
∈
⋂
α
∈
β
C.
α
}
∈
C.
λ
.
{\ displaystyle \ textstyle \ bigtriangleup _ {\ alpha \ in \ lambda} C _ {\ alpha}: = \ lbrace \ beta \ in \ lambda \ mid \ beta \ in \ bigcap _ {\ alpha \ in \ beta} C_ { \ alpha} \ rbrace \ in {\ mathcal {C}} _ {\ lambda}.}
The too dual ideal, defined by , is called the ideal of the thin subsets .
C.
λ
{\ displaystyle {\ mathcal {C}} _ {\ lambda}}
I.
λ
=
{
D.
⊆
λ
∣
λ
∖
D.
∈
C.
λ
}
{\ displaystyle {\ mathcal {I}} _ {\ lambda} = \ {D \ subseteq \ lambda \ mid \ lambda \ setminus D \ in {\ mathcal {C}} _ {\ lambda} \}}
A set is called stationary if it is not thin, so we have. A crowd is stationary precisely when its section is not empty with every club crowd.
S.
⊆
λ
{\ displaystyle S \ subseteq \ lambda}
S.
∉
I.
λ
{\ displaystyle S \ notin {\ mathcal {I}} _ {\ lambda}}
See also
literature
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