In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is a contraction of "closed and unbounded".

Formal definitions

is closed if and only if Thus, if the limit of some sequence

is unbounded

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

Closed proper classes of ordinals are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).

Examples

The set of all countable limit ordinals is a club set with respect to the first uncountable ordinal is an uncountable initial ordinal In fact a club set is nothing else but the range of a normal function (i.e. increasing and continuous).

is a cardinal) is club (see stationary set).

The closed unbounded filter

be a limit ordinal of uncountable cofinality which shows that the intersection is unbounded.

is a regular cardinal, then

-complete proper filter; that is, on the poset.

is a regular cardinal then club sets are also closed under diagonal intersection.

must include all club sets.

See also

References

• Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. .

• Lévy, Azriel (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover.