Zariski topology


Let 𝔸kn denote the affine space kn over a field k. The Zariski topologyMathworldPlanetmath on 𝔸kn is defined to be the topology whose closed sets are the sets

V(I):={x𝔸knf(x)=0 for all fI}𝔸kn,

where Ik[X1,,Xn] is any ideal in the polynomial ring k[X1,,Xn]. For any affine varietyMathworldPlanetmath V𝔸kn, the Zariski topology on V is defined to be the subspace topology induced on V as a subset of 𝔸kn.

Let kn denote n–dimensional projective spaceMathworldPlanetmath over k. The Zariski topology on kn is defined to be the topology whose closed sets are the sets

V(I):={xknf(x)=0 for all fI}kn,

where Ik[X0,,Xn] is any homogeneous idealMathworldPlanetmath in the graded kalgebra k[X0,,Xn]. For any projective variety Vkn, the Zariski topology on V is defined to be the subspace topology induced on V as a subset of kn.

The Zariski topology is the predominant topology used in the study of algebraic geometryMathworldPlanetmathPlanetmath. Every regular morphism of varietiesPlanetmathPlanetmath is continuous in the Zariski topology (but not every continuous map in the Zariski topology is a regular morphism). In fact, the Zariski topology is the weakest topology on varieties making points in 𝔸k1 closed and regular morphisms continuous.

Title Zariski topology
Canonical name ZariskiTopology
Date of creation 2013-03-22 12:38:11
Last modified on 2013-03-22 12:38:11
Owner djao (24)
Last modified by djao (24)
Numerical id 4
Author djao (24)
Entry type Definition
Classification msc 14A10
Related topic PrimeSpectrum