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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 Infinity-Borel set ： ウィキペディア英語版 Infinity-Borel set In set theory, a subset of a Polish space $X$ is ∞-Borel if it can be obtained by starting with the open subsets of $X$, and transfinitely iterating the operations of complementation and wellordered union. Note that the set of ∞-Borel sets may not actually be closed under wellordered union; see below. ==Formal definition== More formally: we define by simultaneous transfinite recursion the notion of ∞-Borel code, and of the interpretation of such codes. Since $X$ is Polish, it has a countable base. Let enumerate that base (that is, $\backslash mathcal\_i$ is the $i^\backslash mathrm$ basic open set). Now: * Every natural number $i$ is an ∞-Borel code. Its interpretation is $\backslash mathcal\_i$. * If $c$ is an ∞-Borel code with interpretation $A\_c$, then the ordered pair $\backslash left\backslash langle\; 0,c\backslash right\backslash rangle$ is also an ∞-Borel code, and its interpretation is the complement of $A\_c$, that is, $X\backslash setminus\; A\_c$. * If $\backslash vec\; c$ is a length-α sequence of ∞-Borel codes for some ordinal α (that is, if for every β<α, $c\_\backslash beta$ is an ∞-Borel code, say with interpretation $A\_A\_\{c\_\{\backslash beta\}\}$. Now a set is ∞-Borel if it is the interpretation of some ∞-Borel code. The axiom of choice implies that ''every'' set can be wellordered, and therefore that every subset of every Polish space is ∞-Borel. Therefore the notion is interesting only in contexts where AC does not hold (or is not known to hold). Unfortunately, without the axiom of choice, it is not clear that the ∞-Borel sets ''are'' closed under wellordered union. This is because, given a wellordered union of ∞-Borel sets, each of the individual sets may have ''many'' ∞-Borel codes, and there may be no way to choose one code for each of the sets, with which to form the code for the union. The assumption that every set of reals is ∞-Borel is part of AD+, an extension of the axiom of determinacy studied by Woodin. 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「Infinity-Borel 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.