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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 Kuratowski's closure-complement problem ： ウィキペディア英語版 Kuratowski's closure-complement problem In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922. The problem gained wide exposure three decades later as an exercise in John L. Kelley's classic textbook ''General Topology''. ==Proof== Letting ''S'' denote an arbitrary subset of a topological space, write ''kS'' for the closure of ''S'', and ''cS'' for the complement of ''S''. The following three identities imply that no more than 14 distinct sets are obtainable: (1) ''kkS'' = ''kS''. (The closure operation is idempotent.) (2) ''ccS'' = ''S''. (The complement operation is an involution.) (3) ''kckckckcS'' = ''kckcS''.(Or equivalently ''kckckckS=kckckckccS=kckS''. Using identity (2).) The first two are trivial. The third follows from the identity ''kikiS'' = ''kiS'' where ''iS'' is the interior of ''S'' which is equal to the complement of the closure of the complement of ''S'', ''iS'' = ''ckcS''. (The operation ''ki'' = ''kckc'' is idempotent.) A subset realizing the maximum of 14 is called a 14-set. The space of real numbers under the usual topology contains 14-sets. Here is one example: :$(0,1)\backslash cup(1,2)\backslash cup\backslash \backslash cup\backslash bigl(()\backslash cap\backslash Q\backslash bigr),$ where $(1,2)$ denotes an open interval and $()$ denotes a closed interval. 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「Kuratowski's closure-complement problem」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.