Club set
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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 limit ordinal, then a set 
is closed in κ if and only if for every α < κ, if 
, then 
. Thus, if the limit of some sequence in C is less than κ, then the limit is also in C.
If κ is a limit ordinal and then C is unbounded in κ if and only if for any α < κ, there is some 
such that α < β.
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.
[edit] See also
[edit] 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
- This article incorporates material from Club on PlanetMath, which is licensed under the GFDL.
