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In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring. The Zariski topology allows using tools of topology for the study of algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows to build general algebraic varieties by gluing together affine varieties in a similar way as it is done in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety. In the case of an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology. The generalization of the Zariski topology to the set of prime ideals of a commutative ring follows from Hilbert's Nullstellensatz that establishes a bijective correspondence between the points of an affine variety defined over an algebraically closed field and the maximal ideals of the ring of its regular functions. This suggests to define the Zariski topology on the set of the maximal ideals of a commutative ring as the topology such that a set of maximal ideals is closed if and only if it is the set of all maximal ideals that contain a given ideal. Another basic idea of Grothendieck's scheme theory is to consider as ''points'', not only the usual points corresponding to maximal ideals, but also all (irreducible) algebraic varieties, which correspond to prime ideals. Thus the Zariski topology on the set of prime ideals (spectrum) of a commutative ring is the topology such that a set of prime ideals is closed if and only it is the set of all prime ideals that contain a fixed ideal. ==Zariski topology of varieties== In classical algebraic geometry (that is, the part of algebraic geometry in which one does not use schemes, that have been introduced by Grothendieck around 1960), the Zariski topology is defined on algebraic varieties. The Zariski topology, defined on the points of the variety, is the topology such that the closed sets are the algebraic subsets of the variety. As the most elementary algebraic varieties are affine and projective varieties, it is useful to make this definition more eplicit in both cases. We assume that we are working over a fixed, algebraically closed field ''k'' (in classical geometry ''k'' is almost always the complex numbers). 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「Zariski topology」の詳細全文を読む スポンサード リンク
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