Zariski Topology

The Zariski topology is a topology that is well-suited for the study of polynomial equations in algebraic geometry, since a Zariski topology has many fewer open sets than in the usual metric topology. In fact, the only closed sets are the algebraic sets, which are the zeros of polynomials.

For example, in C, the only nontrivial closed sets are finite collections of points. In C^2, there are also the zeros of polynomials such as lines ax+by and cusps x^2+y^3.

The Zariski topology is not a T2-space. In fact, any two open sets must intersect, and cannot be disjoint. Also, the open sets are dense, in the Zariski topology as well as in the usual metric topology.

Because there are fewer open sets than in the usual topology, it is more difficult for a function to be continuous in Zariski topology. For example, a continuous function (C^n,Zariski)->(C,metric) must be a constant function. Conversely, when the range has the Zariski topology, it is easier for a function to be continuous. In particular, the polynomials are continuous functions (C^n,Zariski)->(C,Zariski).

In general, the Zariski topology of a ring R is a topology on the set of prime ideals, known as the ring spectrum. Its closed sets are V(a), where a is any ideal in R and V(a) is the set of prime ideals containing a.

See also

Algebraic Variety, Category Theory, Commutative Algebra, Conic Section, Ideal, Prime Ideal, Scheme

This entry contributed by Todd Rowland

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Reid, M. Undergraduate Algebraic Geometry. New York: Cambridge University Press, 1988.

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Zariski Topology

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Rowland, Todd. "Zariski Topology." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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