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 a x + b y and cusps x^2 + y^3. The Zariski topology is not a T_2-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.

algebraic variety | category theory | commutative algebra | conic section | ideal | prime ideal | scheme