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. ${\displaystyle S\subseteq \lambda }$ ${\displaystyle S\notin {\mathcal {I}}_{\lambda }}$

See also

• Fodor's theorem
• Reflection principle (set theory)

literature

• Thomas Jech : Set Theory. 3rd millenium edition, revised and expanded, corrected 4th print. Springer, Berlin et al. 2006, ISBN 3-540-44085-2 .