club crowd

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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).


Let be a Limesordinal number. A subset is called

  • complete if for any sequence of the following applies:
  • unlimited if all one does with .

is called the club set if it is both completed and unlimited.


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.

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.

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:

  • is -complete: Is and for , then applies
  • If a regular cardinal number is , then so -called diagonal cuts are closed : If a family is made up of club sets , so is

The too dual ideal, defined by , is called the ideal of the thin subsets .

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.

See also