Zariski topology

plural Zariski topologies

1. algebraic geometryOriginally, a topology applicable to algebraic varieties, such that the closed sets are the variety's algebraic subvarieties; later, a generalisation in which the topological space is the set of prime ideals of a commutative ring and is called the spectrum of the ring.

   The Zariski topology allows tools from topology to be used in the study of algebraic varieties, even when the underlying field is not a topological field.

   3 more examples

   Thus the Zariski open subsets are 'very big'; in particular, the Zariski topology is highly non-Hausdorff. A further difference with the classical topology is that the Zariski topology on the product X#92;timesY of two affine varieties is stronger than the product of the Zariski topologies on X and Y.

   1994, V. I. Danilov, “II. Algebraic Varieties and Schemes”, in I. R. Shafarevich, editor, Algebraic Geometry I: Algebraic Curves, Algebraic Manifolds and Schemes, Springer, page 184

   In particular, “singular” is a local notion, where “local” so far was mainly considered with respect to the Zariski topology. However, since the Zariski topology is so coarse, small neighbourhoods in the Zariski topology might not be local enough.

   2008, Gert-Martin Greuel, Gerhard Pfister, A Singular Introduction to Commutative Algebra, 2nd edition, Springer, page 523

   If Y#92;subseteqKⁿ is an affine variety, then by definition the Zariski topology on Y has the subvarieties of Y as closed sets.

   2011, Gregor Kemper, A Course in Commutative Algebra, Springer, page 34