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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 of ''A'' in ''X'' is the union of ''A'' and the set of all limits of sequences of elements in ''A'' (its ''limit points''), : Then ''A'' is dense in ''X'' if : Note that . If is a sequence of dense open sets in a complete metric space, ''X'', then 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
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