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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 Metric space aimed at its subspace ： ウィキペディア英語版 Metric space aimed at its subspace In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the ''metric envelope'', or tight span, which are basic (injective) objects of the category of metric spaces. Following , a notion of a metric space ''Y'' aimed at its subspace ''X'' is defined. Informally, imagine terrain Y, and its part X, such that wherever in Y you place a sharpshooter, and an apple at another place in Y, and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of X, or at least it will fly arbitrarily close to points of X – then we say that Y is aimed at X. A priori, it may seem plausible that for a given ''X'' the superspaces ''Y'' that aim at ''X'' can be arbitrarily large or at least huge. We will see that this is not the case. Among the spaces, which aim at a subspace isometric to ''X'' there is a unique (up to isometry) universal one, Aim(''X''), which in a sense of canonical isometric embeddings contains any other space aimed at (an isometric image of) ''X''. And in the special case of an arbitrary compact metric space ''X'' every bounded subspace of an arbitrary metric space ''Y'' aimed at ''X'' is totally bounded (i.e. its metric completion is compact). == Definitions == Let $(Y,\; d)$ be a metric space. Let $X$ be a subset of $Y$, so that $(X,d\; |\_X)$ (the set $X$ with the metric from $Y$ restricted to $X$) is a metric subspace of $(Y,d)$. Then Definition.  Space $Y$ aims at $X$ if and only if, for all points $y,\; z$ of $Y$, and for every real $\backslash epsilon\; >\; 0$, there exists a point $p$ of $X$ such that :$|d(p,y)\; -\; d(p,z)|\; >\; d(y,z)\; -\; \backslash epsilon.$ Let $\backslash text(X)$ be the space of all real valued metric maps (non-contractive) of $X$. Define :$\backslash text(X)\; :=\; \backslash \; p,q\backslash in\; X\backslash \}.$ Then : for every $f,\; g\backslash in\; \backslash text(X)$ is a metric on $\backslash text(X)$. Furthermore, $\backslash delta\_X\backslash colon\; x\backslash mapsto\; d\_x$, where $d\_x(p)\; :=\; d(x,p)\backslash ,$, is an isometric embedding of $X$ into $\backslash operatorname(X)$; this is essentially a generalisation of the Kuratowski-Wojdysławski embedding of bounded metric spaces $X$ into $C(X)$, where we here consider arbitrary metric spaces (bounded or unbounded). It is clear that the space $\backslash operatorname(X)$ is aimed at $\backslash delta\_X(X)$. 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「Metric space aimed at its subspace」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.