# 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}}$