Definitions for zariski topology
zaris·ki topol·o·gy

WiktionaryRate this definition:0.0 / 0 votes

1. Zariski topologynoun

   A particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition.

WikipediaRate this definition:0.0 / 0 votes

1. Zariski topology

   In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that 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 defining 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 if it is the set of all prime ideals that contain a fixed ideal.

WikidataRate this definition:0.0 / 0 votes

1. Zariski topology

   In algebraic geometry, the Zariski topology is a topology chosen for algebraic varieties. It is due to Oscar Zariski and took a place of particular importance in the field around 1950. The more subtle étale topology is a refinement of the Zariski topology discovered by Grothendieck in the 1960s.

How to pronounce zariski topology?

1. Alex

   US English

   David

   US English

   Mark

   US English

   Daniel

   British

   Libby

   British

   Mia

   British

   Karen

   Australian

   Hayley

   Australian

   Natasha

   Australian

   Veena

   Indian

   Priya

   Indian

   Neerja

   Indian

   Zira

   US English

   Oliver

   British

   Wendy

   British

   Fred

   US English

   Tessa

   South African

How to say zariski topology in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of zariski topology in Chaldean Numerology is: 3

2. Pythagorean Numerology

   The numerical value of zariski topology in Pythagorean Numerology is: 2

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Zariski_Topology
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Zariski_Topology
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=zariski topology