Meaning of Zariski topology | Babel Free

Definitions

Originally, 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.

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Examples

“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.”

“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.”

“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.”

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

CEFR level