In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that 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.
Zariski topology of varieties In classical algebraic geometry (that is, the part of algebraic geometry in which one does not use schemes, which were introduced by Grothendieck around 1960), the Zariski topology is defined on algebraic varieties. The Zariski topology, defined on the points of the variety, is the topology such that the closed sets are the algebraic subsets of the variety. As the most elementary algebraic varieties are affine and projective varieties, it is useful to make this definition more explicit in both cases. We assume that we are working over a fixed, algebraically closed field k (in classical algebraic geometry, k is usually the field of complex numbers).
Affine varieties First, we define the topology on the affine space A n {\displaystyle \mathbb {A} ^{n}} , formed by the n {\displaystyle n} -tuples of elements of k {\displaystyle k} . The topology is defined by specifying its closed sets, rather than its open sets, and these are taken simply to be all the algebraic sets in A n {\displaystyle \mathbb {A} ^{n}} . That is, the closed sets are those of the form
V ( S ) = { x ∈ A n ∣ f ( x ) = 0 , ∀ f ∈ S } {\displaystyle V(S)=\{x\in \mathbb {A} ^{n}\mid f(x)=0,\quad \forall f\in S\}}
where S {\displaystyle S} is any set of polynomials in n {\displaystyle n} variables over k {\displaystyle k} . It is a straightforward verification to show that:
V ( S ) = V ( ( S ) ) {\displaystyle V(S)=V((S))} , where ( S ) {\displaystyle (S)} is the ideal generated by the elements of S {\displaystyle S} ; For any two ideals of polynomials I , J {\displaystyle I,J} , we have
V ( I ) ∪ V ( J ) = V ( I J ) ; {\displaystyle V(I)\cup V(J)=V(IJ);}
V ( I ) ∩ V ( J ) = V ( I + J ) . {\displaystyle V(I)\cap V(J)=V(I+J).}
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