I am completely new to algebraic geometry so please bear with me. I have started going through James Milne's notes as a first reference and have been finding them quite good. I have now turned to Hartshorne's book to get a basic idea of the definition of a scheme and I am confused by the different approaches. In particular, Milne seems to define the Zariski topology on the affine space, such as R^n or C^n. However, Hartshorne, as well as most other authors, define it to be a topology on the set Spec(A). What is the connection between these? My first thought is that they are homemorphic topological spaces since the algebraic sets correspond to radical ideals, but then why isn't Spec(A) defined as the set of radical ideals rather than the set of prime ideals? I know the question is somewhat vague, but that is because my understanding is very vague, but in short, what is the connection between defining a topology where the closed sets are algebraic subsets of the affine space, and defining a topology where the closed sets are the set of prime ideals which contain some other ideal?
Thanks
You are asking a good question in the sense that its answer will give you a better understanding of modern algebraic geometry: the topological space one gets when taking affine space $\mathbb{A}^n(k)$ over an algebraically closed field $k$ equipped with the Zariski topology is homeomorphic to the topological subspace of $\mathrm{Spec}(k[x_1,\ldots ,x_n])$ (equipped with the Zariski topolgy) consisting of the closed points only. This is one of the first basic results in algebraic geometry and it relates the "classical approach" to modern scheme theory. Essentially it means that the two approaches are equivalent.