翻訳と辞書 Words near each other ・ Lebesby Church ・ Lebesgue (crater) ・ Lebesgue constant ・ Lebesgue constant (interpolation) ・ Lebesgue covering dimension ・ Lebesgue differentiation theorem ・ Lebesgue integrability ・ Lebesgue integration ・ Lebesgue measure ・ Lebesgue point ・ Lebesgue space ・ Lebesgue spine ・ Lebesgue's decomposition theorem ・ Lebesgue's density theorem ・ Lebesgue's lemma ・ Lebesgue's number lemma ・ Lebesgue–Stieltjes integration ・ Lebesque ・ Lebet ・ Lebetain ・ Lebetus ・ Lebeuville ・ Lebewohl, Fremde ・ Lebhar IMP Pairs ・ Lebia ・ Lebia analis ・ Lebia chlorocephala ・ Lebia cruxminor ・ Lebia cyanocephala ・ Lebia darlingtoniana Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lebesgue's number lemma ： ウィキペディア英語版 Lebesgue's number lemma In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states: :If the metric space $(X,\; d)$ is compact and an open cover of $X$ is given, then there exists a number $\backslash delta\; >\; 0$ such that every subset of $X$ having diameter less than $\backslash delta$; is contained in some member of the cover. Such a number $\backslash delta$ is called a Lebesgue number of this cover. The notion of a Lebesgue number itself is useful in other applications as well. == Proof == Let $\backslash mathcal\; U$ be an open cover of $X$. Since $X$ is compact we can extract a finite subcover $\backslash \; \backslash subseteq\; \backslash mathcal\; U$. For each $i\; \backslash in\; \backslash$, let $C\_i\; :=\; X\; \backslash setminus\; A\_i$ and define a function $f\; :\; X\; \backslash rightarrow\; \backslash mathbb\; R$ by $f(x)\; :=\; \backslash frac\; \backslash sum\_^n\; d(x,C\_i)$. Since $f$ is continuous on a compact set, it attains a minimum $\backslash delta$. The key observation is that $\backslash delta\; >\; 0$. If $Y$ is a subset of $X$ of diameter less than $\backslash delta$, then there exist $x\_0\backslash in\; X$ such that $Y\backslash subseteq\; B\_\backslash delta(x\_0)$, where $B\_\backslash delta(x\_0)$ denotes the radius $\backslash delta$ ball centered at $x\_0$ (namely, one can choose as $x\_0$ any point in $Y$). Since $f(x\_0)\backslash geq\; \backslash delta$ there must exist at least one $i$ such that $d(x\_0,C\_i)\backslash geq\; \backslash delta$. But this means that $B\_\backslash delta(x\_0)\backslash subseteq\; A\_i$ and so, in particular, $Y\backslash subseteq\; A\_i$. 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「Lebesgue's number lemma」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.