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In mathematics, a hyperconnected space is a topological space ''X'' that cannot be written as the union of two non-empty closed sets (whether disjoint or non-disjoint). The name irreducible space is preferred in algebraic geometry. For a topological space ''X'' the following conditions are equivalent: * No two nonempty open sets are disjoint. * ''X'' cannot be written as the union of two proper closed sets. * Every nonempty open set is dense in ''X''. * The interior of every proper closed set is empty. A space which satisfies any one of these conditions is called ''hyperconnected'' or ''irreducible''. An irreducible set is a subset of a topological space for which the subspace topology is irreducible. Some authors do not consider the empty set to be irreducible (even though it vacuously satisfies the above conditions). == Examples == Examples of hyperconnected spaces include the cofinite topology on any infinite space and the Zariski topology on an algebraic variety. 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「Hyperconnected space」の詳細全文を読む スポンサード リンク
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