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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Stationary set ： ウィキペディア英語版 Stationary set In mathematical set theory and model theory, a stationary set is one that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in set theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset. ==Classical notion== If $\backslash kappa\; \backslash ,$ is a cardinal of uncountable cofinality, $S\; \backslash subseteq\; \backslash kappa\; \backslash ,,$ and $S\; \backslash ,$ intersects every club set in $\backslash kappa\; \backslash ,,$ then $S\; \backslash ,$ is called a stationary set.〔Jech (2003) p.91〕 If a set is not stationary, then it is called a thin set. This notion should not be confused with the notion of a thin set in number theory. If $S\; \backslash ,$ is a stationary set and $C\; \backslash ,$ is a club set, then their intersection $S\; \backslash cap\; C\; \backslash ,$ is also stationary. This is because if $D\; \backslash ,$ is any club set, then $C\; \backslash cap\; D\; \backslash ,$ is a club set, thus $(S\; \backslash cap\; C)\; \backslash cap\; D\; =\; S\; \backslash cap\; (C\; \backslash cap\; D)\; \backslash ,$ is non empty. Therefore, $(S\; \backslash cap\; C)\; \backslash ,$ must be stationary. ''See also'': Fodor's lemma The restriction to uncountable cofinality is in order to avoid trivialities: Suppose $\backslash kappa$ has countable cofinality. Then $S\backslash subset\backslash kappa$ is stationary in $\backslash kappa$ if and only if $\backslash kappa\backslash setminus\; S$ is bounded in $\backslash kappa$. In particular, if the cofinality of $\backslash kappa$ is $\backslash omega=\backslash aleph\_0$, then any two stationary subsets of $\backslash kappa$ have stationary intersection. This is no longer the case if the cofinality of $\backslash kappa$ is uncountable. In fact, suppose $\backslash kappa$ is regular and $S\backslash subset\backslash kappa$ is stationary. Then $S$ can be partitioned into $\backslash kappa$ many disjoint stationary sets. This result is due to Solovay. If $\backslash kappa$ is a successor cardinal, this result is due to Ulam and is easily shown by means of what is called an Ulam matrix. H. Friedman has shown that for every countable successor ordinal $\backslash beta$, every stationary subset of $\backslash omega\_1$ contains a closed subset of order type $\backslash beta$ (Friedman). 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「Stationary set」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.