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

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

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.

This article incorporates material from Club on PlanetMath, which is licensed under the GFDL.