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Zariski topology

Zariski topology noun: 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.

Noun

Zariski topology (plural Zariski topologies)

  1. 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.
    • 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.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

Examples of Zariski topology in a sentence

  • In particular, “singular” is a local notion, where “local” so far was mainly considered with respect to the Zariski topology.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:

Forms of Zariski topology

pluralZariski topologies

Origin of Zariski topology

After Russian-born American mathematician Oscar Zariski (1899—1986), regarded as one of the most influential algebraic geometers of the 20th century.

Questions about Zariski topology

What does Zariski topology mean?
Zariski topology means 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.
How do you use Zariski topology in a sentence?
In particular, “singular” is a local notion, where “local” so far was mainly considered with respect to the Zariski topology.
What part of speech is Zariski topology?
Zariski topology is a noun.
What is the plural of Zariski topology?
The plural of Zariski topology is Zariski topologies.

Words near Zariski topology in the dictionary

  1. zaris
  2. Zariski
  3. Zariski space
  4. Zariski spaces
  5. Zariski topologies
  6. Zariski topology
  7. Zariski-Riemann space
  8. Zariski-Riemann spaces
  9. Zariskian
  10. Zariyah
  11. Zarjeh Bostan

Source and license

Adapted from Wiktionary: Zariski topology (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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