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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 Star refinement ： ウィキペディア英語版 Star refinement In mathematics, specifically in the study of topology and open covers of a topological space ''X'', a star refinement is a particular kind of refinement of an open cover of ''X''. The general definition makes sense for arbitrary coverings and does not require a topology. Let $X$ be a set and let $\backslash mathcal\; U=(U\_i)\_$ be a covering of $X$, i.e., $X=\backslash bigcup\_U\_i$. Given a subset $S$ of $X$ then the ''star'' of $S$ with respect to $\backslash mathcal\; U$ is the union of all the sets $U\_i$ that intersect $S$, i.e.: : $\backslash mathrm(S,\backslash mathcal\; U)=\backslash bigcup\backslash big\backslash .$ Given a point $x\backslash in\; X$, we write $\backslash mathrm(x,\backslash mathcal\; U)$ instead of $\backslash mathrm(\backslash ,\backslash mathcal\; U)$. The covering $\backslash mathcal\; U=(U\_i)\_$ of $X$ is said to be a ''refinement'' of a covering $\backslash mathcal\; V=(V\_j)\_$ of $X$ if every $U\_i$ is contained in some $V\_j$. The covering $\backslash mathcal\; U$ is said to be a ''barycentric refinement'' of $\backslash mathcal\; V$ if for every $x\backslash in\; X$ the star $\backslash mathrm(x,\backslash mathcal\; U)$ is contained in some $V\_j$. Finally, the covering $\backslash mathcal\; U$ is said to be a ''star refinement'' of $\backslash mathcal\; V$ if for every $i\backslash in\; I$ the star $\backslash mathrm(U\_i,\backslash mathcal\; U)$ is contained in some $V\_j$. Star refinements are used in the definition of fully normal space and in one definition of uniform space. It is also useful for stating a characterization of paracompactness. == References == * J. Dugundji, Topology, Allyn and Bacon Inc., 1966. * Lynn Arthur Steen and J. Arthur Seebach, Jr.; 1970; ''Counterexamples in Topology''; 2nd (1995) Dover edition ISBN 0-486-68735-X; page 165. 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「Star refinement」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.