# club

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 $C$ is less than $\kappa$ then the limit is also in $C$.)

If $\kappa$ is a cardinal and $C\subseteq\kappa$ then $C$ 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.

 Title club Canonical name Club Date of creation 2013-03-22 12:53:01 Last modified on 2013-03-22 12:53:01 Owner Henry (455) Last modified by Henry (455) Numerical id 5 Author Henry (455) Entry type Definition Classification msc 03E10 Defines club Defines closed Defines unbounded Defines closed unbounded Defines closed set Defines unbounded set Defines closed unbounded set Defines club set