Science Fair Project Encyclopedia

Science Fair Project Encyclopedia Contents Page

Club set

In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded.

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

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

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

For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor bounded.

Categories: Set theory

Science Fair Project Encyclopedia Contents Page

10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details

All Science Fair Projects

Science Fair Project Encyclopedia for Schools!

Science Fair Project Encyclopedia

Club set