翻訳と辞書 Words near each other ・ Denscantia ・ Dense (film) ・ Dense artery sign ・ Dense bodies ・ Dense connective tissue ・ Dense granule ・ Dense graph ・ Dense heterarchy ・ Dense Inert Metal Explosive ・ Dense irregular connective tissue ・ Dense non-aqueous phase liquid ・ Dense order ・ Dense Pack ・ Dense plasma focus ・ Dense regular connective tissue ・ Dense set ・ Dense subgraph ・ Dense submodule ・ Dense Time ・ Dense-in-itself ・ Dense-rock equivalent ・ Dense-scale lantern shark ・ Denseisha ・ Densely defined operator ・ Densely packed decimal ・ Densen Audio Technologies ・ Densen Uta ・ Densetsu no Shōjo ・ Densetsu no Stafy (video game) ・ Densetsu no Stafy 2 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 Dense set ： ウィキペディア英語版 Dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is called dense (in ''X'') if every point ''x'' in ''X'' either belongs to ''A'' or is a limit point of ''A''. Informally, for every point in ''X'', the point is either in ''A'' or arbitrarily "close" to a member of ''A'' — for instance, every real number is either a rational number or has one arbitrarily close to it (see Diophantine approximation). Formally, a subset ''A'' of a topological space ''X'' is dense in ''X'' if for any point ''x'' in ''X'', any neighborhood of ''x'' contains at least one point from ''A'' (i.e., ''A'' has non-empty intersection with every non-empty open subset of ''X''). Equivalently, ''A'' is dense in ''X'' if and only if the only closed subset of ''X'' containing ''A'' is ''X'' itself. This can also be expressed by saying that the closure of ''A'' is ''X'', or that the interior of the complement of ''A'' is empty. The density of a topological space ''X'' is the least cardinality of a dense subset of ''X''. ==Density in metric spaces== An alternative definition of dense set in the case of metric spaces is the following. When the topology of ''X'' is given by a metric, the closure $\backslash displaystyle\; \backslash overline$ of ''A'' in ''X'' is the union of ''A'' and the set of all limits of sequences of elements in ''A'' (its ''limit points''), :$\backslash overline\; =\; A\; \backslash cup\; \backslash$ Then ''A'' is dense in ''X'' if :$\backslash overline\; =\; X$ Note that $\backslash displaystyle\; A\; \backslash ;\backslash subseteq\backslash ;\; \backslash$. If $\backslash displaystyle\; \backslash$ is a sequence of dense open sets in a complete metric space, ''X'', then $\backslash displaystyle\; \backslash bigcap^\_\; U\_n$ is also dense in ''X''. This fact is one of the equivalent forms of the Baire category theorem. 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「Dense set」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.