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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 Separating set ： ウィキペディア英語版 Separating set In mathematics a set of functions ''S'' from a set ''D'' to a set ''C'' is called a separating set for ''D'' or said to separate the points of ''D'' if for any two distinct elements ''x'' and ''y'' of ''D'', there exists a function ''f'' in ''S'' so that ''f''(''x'') ≠ ''f''(''y'').〔.〕 Separating sets can be used to formulate a version of the Stone-Weierstrass theorem for real-valued functions on a compact Hausdorff space ''X'', with the topology of uniform convergence. It states that any subalgebra of this space of functions is dense if and only if it separates points. This is the version of the theorem originally proved by Marshall H. Stone.〔 == Examples == * The singleton set consisting of the identity function on R separates the points of R. * If ''X'' is a T1 normal topological space, then Urysohn's lemma states that the set C(''X'') of continuous functions on ''X'' with real (or complex) values separates points on ''X''. 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「Separating 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.