Definitions

Club_set

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 limit ordinal, then a set Csubseteqkappa is closed in kappa if and only if for every alpha, if sup(Ccap alpha)=alphane0, then alphain C. Thus, if the limit of some sequence in C is less than kappa, then the limit is also in C.

If kappa is a limit ordinal and Csubseteqkappa then C is unbounded in kappa if and only if for any alpha, there is some betain C such that alpha.

If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).

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 unbounded.

See also

References

  • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2.
  • Levy, A. (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover. ISBN 0-486-42079-5
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