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

\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. 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.

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