Formal Definition
Formally, if is a limit ordinal, then a set is closed in if and only if for every, if, then . Thus, if the limit of some sequence in is less than, then the limit is also in .
If is a limit ordinal and then is unbounded in if and only if for any, there is some such that .
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. The set of all limit ordinals is closed unbounded in ( regular). In fact a club set is nothing else but the range of a normal function (i.e. increasing and continuous).
Read more about this topic: Club Set
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