Stationary Set - Classical Notion

Classical Notion

If is a cardinal of uncountable cofinality, and intersects every club set in then is called a stationary set. If a set is not stationary, then it is called a thin set. This notion should not be confused with the notion of a thin set in number theory.

If is a stationary set and is a club set, then their intersection is also stationary. Because if is any club set, then is a club set because the intersection of two club sets is club. Thus is non empty. Therefore must be stationary.

See also: Fodor's lemma

The restriction to uncountable cofinality is in order to avoid trivialities: Suppose has countable cofinality. Then is stationary in if and only if is bounded in . In particular, if the cofinality of is, then any two stationary subsets of have stationary intersection.

This is no longer the case if the cofinality of is uncountable. In fact, suppose is regular and is stationary. Then can be partitioned into many disjoint stationary sets. This result is due to Solovay. If is a successor cardinal, this result is due to Ulam and is easily shown by means of what is called an Ulam matrix.

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