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In mathematics, and specifically in algebraic geometry, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation :''XY'' = 0 is the union of the two lines :''X'' = 0 and :''Y'' = 0. Thus an algebraic set is irreducible if it is not the union of two proper algebraic subsets. It is a fundamental theorem of classical algebraic geometry that every algebraic set is the union of a finite number of irreducible algebraic subsets (varieties) and that this decomposition is unique if one removes those subsets that are contained in another one. The elements of this unique decomposition are called irreducible components. This notion may be reformulated in topological terms, using Zariski topology, for which the closed sets are the subvarieties: an algebraic set is irreducible if it is not the union of two proper subsets that are closed for Zariski topology. This allows a generalization in topology, and, through it, to general schemes for which the above property of finite decomposition is not necessarily true. == In topology == A topological space ''X'' is reducible if it can be written as a union of two non-empty closed proper subsets , of . A topological space is irreducible (or hyperconnected) if it is not reducible. Equivalently, all non empty open subsets of ''X'' are dense or any two nonempty open sets have nonempty intersection. A subset ''F'' of a topological space ''X'' is called irreducible or reducible, if ''F'' considered as a topological space via the subspace topology has the corresponding property in the above sense. That is, is reducible if it can be written as a union where are closed subsets of , neither of which contains . An irreducible component of a topological space is a maximal irreducible subset. If a subset is irreducible, its closure is, so irreducible components are closed. 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「irreducible component」の詳細全文を読む スポンサード リンク
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