PlanetMath: club

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club

(Definition)

If $\kappa$ is a cardinal then a set $C\subseteq\kappa$ is closed iff for any $S\subseteq C$ and $\alpha<\kappa$ , $\sup(S\cap \alpha)=\alpha$ then $\alpha\in C$ . (That is, if the limit of some sequence in is less than $\kappa$ then the limit is also in .)

If $\kappa$ is a cardinal and $C\subseteq\kappa$ then is unbounded if, for any $\alpha<\kappa$ , there is some $\beta\in C$ such that $\alpha<\beta$ .

If a set is both closed and unbounded then it is a club set.

"club" is owned by Henry.

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Also defines:

club, closed, unbounded, closed unbounded, closed set, unbounded set, closed unbounded set, club set

Attachments:: club filter (Definition) by Henry

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Cross-references: sequence, limit, iff, cardinal
There are 119 references to this entry.

This is version 2 of club, born on 2002-07-29, modified 2008-02-15.
Object id is 3227, canonical name is Club.
Accessed 14944 times total.

Classification:

AMS MSC:

03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)

Pending Errata and Addenda

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