Let 𝔸kn denote the affine space kn over a field k. The Zariski topology
on 𝔸kn is defined to be the topology whose closed sets are the sets
| V(I):={x∈𝔸kn∣f(x)=0 for all f∈I}⊂𝔸kn, |
where I⊂k[X1,…,Xn] is any ideal in the polynomial ring k[X1,…,Xn]. For any affine variety V⊂𝔸kn, the Zariski topology on V is defined to be the subspace topology induced on V as a subset of 𝔸kn.
Let ℙkn denote n–dimensional projective space over k. The Zariski topology on ℙkn is defined to be the topology whose closed sets are the sets
| V(I):={x∈ℙkn∣f(x)=0 for all f∈I}⊂ℙkn, |
where I⊂k[X0,…,Xn] is any homogeneous ideal in the graded k–algebra k[X0,…,Xn]. For any projective variety V⊂ℙkn, the Zariski topology on V is defined to be the subspace topology induced on V as a subset of ℙkn.