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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 (see below) relative to the limit ordinal. The name ''club'' is a contraction of "closed and unbounded". == 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 from is less than , then the limit is also in . If is a limit ordinal and then is unbounded in 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). More generally, if is a nonempty set and is a cardinal, then is ''club'' if every union of a subset of is in and every subset of of cardinality less than is contained in some element of (see stationary set). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「club set」の詳細全文を読む スポンサード リンク
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