The Classical Definition
In classical algebraic geometry (that is, the subject prior to the Grothendieck revolution of the late 1950s and 1960s) the Zariski topology was defined in the following way. Just as the subject itself was divided into the study of affine and projective varieties (see the Algebraic variety definitions) the Zariski topology is defined slightly differently for these two. We assume that we are working over a fixed, algebraically closed field k, which in classical geometry was almost always the complex numbers.
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