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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 discrete space ： ウィキペディア英語版 discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a ''discontinuous sequence'', meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set, i.e., it defines all subsets as open sets. In particular, each singleton is an open set in the discrete topology. == Definitions == Given a set ''X'': * the discrete topology on ''X'' is defined by letting every subset of ''X'' be open (and hence also closed), and ''X'' is a discrete topological space if it is equipped with its discrete topology; * the discrete uniformity on ''X'' is defined by letting every superset of the diagonal in ''X'' × ''X'' be an entourage, and ''X'' is a discrete uniform space if it is equipped with its discrete uniformity. * the discrete metric $\backslash rho$ on ''X'' is defined by :$\backslash rho(x,y)\; =$ \left\\ x\neq y , \\ 0 &\mbox\ x = y \end\right. for any $x,y\; \backslash in\; X$. In this case $(X,\backslash rho)$ is called a discrete metric space or a space of isolated points. * a set ''S'' is discrete in a metric space $(X,d)$, for $S\; \backslash subseteq\; X$, if for every $x\; \backslash in\; S$, there exists some $\backslash delta\; >0$ (depending on $x$) such that $d(x,y)\; >\backslash delta$ for all $y\; \backslash in\; S\backslash setminus\backslash$; such a set consists of isolated points. A set ''S'' is uniformly discrete in the metric space $(X,d)$, for $S\; \backslash subseteq\; X$, if there exists ''ε'' > 0 such that for any two distinct $x,\; y\; \backslash in\; S$, $d(x,\; y)$ > ''ε''. A metric space $(E,d)$ is said to be ''uniformly discrete'' if there exists a "packing radius" $r>0$ such that, for any $x,y\; \backslash in\; E$, one has either $x=y$ or $d(x,y)>r$. The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set of real numbers. Let X = , consider this set using the usual metric on the real numbers. Then, X is a discrete space, since for each point 1/2n, we can surround it with the interval (1/2n - ɛ, 1/2n + ɛ), where ɛ = 1/2(1/2n - 1/2n+1) = 1/2n+2. The intersection (1/2n - ɛ, 1/2n + ɛ) ∩ is just the singleton . Since the intersection of two open sets is open, and singletons are open, it follows that X is a discrete space. However, X cannot be uniformly discrete. To see why, suppose there exists an r>0 such that d(x,y)>r whenever x≠y. It suffices to show that there are at least two points x and y in X that are closer to each other than r. Since the distance between adjacent points 1/2n and 1/2n+1 is 1/2n+1, we need to find an n that satisfies this inequality: $$ 1/2^ < r $$ 1 < r2^ $$ 1/r < 2^ $$ \log_2(1/r) < n+1 $$ -\log_2(r) < n+1 $$ -1 - \log_2(r) < n Since there is always an n bigger than any given real number, it follows that there will always be at least two points in X that are closer to each other than any positive r, therefore X is not uniformly discrete. ... 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「discrete space」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.