# club crowd

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 ${\ displaystyle \ lambda}$ ${\ displaystyle x \ subseteq \ lambda}$ • complete if for any sequence of the following applies: ${\ displaystyle \ langle \ alpha _ {\ xi} \ in x \ mid \ xi <\ mu \ rangle}$ ${\ displaystyle x}$ ${\ displaystyle \ lim _ {\ xi \ to \ mu} \ alpha _ {\ xi} = \ delta \ in \ lambda \ Rightarrow \ delta \ in x,}$ • unlimited if all one does with .${\ displaystyle \ alpha \ in \ lambda}$ ${\ displaystyle \ beta \ in x}$ ${\ displaystyle \ alpha \ leq \ beta}$ ${\ displaystyle x}$ is called the club set if it is both completed and unlimited. ${\ displaystyle x}$ ## 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. ${\ displaystyle \ lambda = \ omega}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ omega}$ 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. ${\ displaystyle \ lambda}$ ${\ displaystyle \ operatorname {Ord}}$ ${\ displaystyle f \ colon \ lambda \ to \ operatorname {Ord}}$ ## 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: ${\ displaystyle \ lambda}$ ${\ displaystyle \ operatorname {cf} \ lambda> \ omega}$ ${\ displaystyle {\ mathcal {C}} _ ​​{\ lambda} = \ {x \ subseteq \ lambda \ mid \ exists C \ subseteq x \ C {\ text {club}} \}}$ ${\ displaystyle {\ mathcal {C _ {\ lambda}}}}$ • ${\ displaystyle {\ mathcal {C}} _ ​​{\ lambda}}$ is -complete: Is and for , then applies ${\ displaystyle \ operatorname {cf} \ lambda}$ ${\ displaystyle \ gamma \ in \ operatorname {cf} \ lambda}$ ${\ displaystyle C _ {\ alpha} \ in {\ mathcal {C}} _ ​​{\ lambda}}$ ${\ displaystyle \ alpha \ in \ gamma}$ ${\ displaystyle \ textstyle \ bigcap \ limits _ {\ alpha \ in \ gamma} C _ {\ alpha} \ in {\ mathcal {C}} _ ​​{\ lambda}.}$ • If a regular cardinal number is , then so -called diagonal cuts are closed : If a family is made up of club sets , so is ${\ displaystyle \ lambda}$ ${\ displaystyle {\ mathcal {C}} _ ​​{\ lambda}}$ ${\ displaystyle \ langle C _ {\ alpha} \ mid \ alpha \ in \ lambda \ rangle}$ ${\ displaystyle {\ mathcal {C}} _ ​​{\ lambda}}$ ${\ displaystyle \ textstyle \ bigtriangleup _ {\ alpha \ in \ lambda} C _ {\ alpha}: = \ lbrace \ beta \ in \ lambda \ mid \ beta \ in \ bigcap _ {\ alpha \ in \ beta} C_ { \ alpha} \ rbrace \ in {\ mathcal {C}} _ ​​{\ lambda}.}$ The too dual ideal, defined by , is called the ideal of the thin subsets . ${\ displaystyle {\ mathcal {C}} _ ​​{\ lambda}}$ ${\ displaystyle {\ mathcal {I}} _ {\ lambda} = \ {D \ subseteq \ lambda \ mid \ lambda \ setminus D \ in {\ mathcal {C}} _ ​​{\ lambda} \}}$ 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. ${\ displaystyle S \ subseteq \ lambda}$ ${\ displaystyle S \ notin {\ mathcal {I}} _ {\ lambda}}$ 