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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 convergence ： ウィキペディア英語版 Kuratowski convergence In mathematics, Kuratowski convergence is a notion of convergence for sequences (or, more generally, nets) of compact subsets of metric spaces, named after Kazimierz Kuratowski. Intuitively, the Kuratowski limit of a sequence of sets is where the sets "accumulate". ==Definitions== Let (''X'', ''d'') be a metric space, where ''X'' is a set and ''d'' is the function of distance between points of ''X''. For any point ''x'' ∈ ''X'' and any non-empty compact subset ''A'' ⊆ ''X'', define the distance between the point and the subset: :$d(x,\; A)\; =\; \backslash inf\; \backslash$. For any sequence of such subsets ''A''''n'' ⊆ ''X'', ''n'' ∈ N, the Kuratowski limit inferior (or lower closed limit) of ''A''''n'' as ''n'' → ∞ is :$\backslash mathop\; A\_\; =\; \backslash left\backslash )\; =\; 0\; \backslash right.\; \backslash right\backslash \}$ ::$=\; \backslash left\backslash \; U\; \backslash mbox\; x,\; \backslash \backslash \; U\; \backslash cap\; A\_\; \backslash neq\; \backslash emptyset\; \backslash mbox\; n\; \backslash end\; \backslash right.\; \backslash right\backslash \};$ the Kuratowski limit superior (or upper closed limit) of ''A''''n'' as ''n'' → ∞ is :$\backslash mathop\; A\_\; =\; \backslash left\backslash )\; =\; 0\; \backslash right.\; \backslash right\backslash \}$ ::$=\; \backslash left\backslash \; U\; \backslash mbox\; x,\; \backslash \backslash \; U\; \backslash cap\; A\_\; \backslash neq\; \backslash emptyset\; \backslash mbox\; n\; \backslash end\; \backslash right.\; \backslash right\backslash \}.$ If the Kuratowski limits inferior and superior agree (i.e. are the same subset of ''X''), then their common value is called the Kuratowski limit of the sets ''A''''n'' as ''n'' → ∞ and denoted Lt''n''→∞''A''''n''. The definitions for a general net of compact subsets of ''X'' go through ''mutatis mutandis''. 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 convergence」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.