one only considers the points of A, i.e. the maximal ideals in R, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets). Specifically, the maximal ideals in R, i.e. , together with the Zariski topology, is homeomorphic to A also with the Zariski topology. One can thus view the topological space as an "enrichment" of the topological space A (with Zariski topology): for every subvariety of A, one additional non-closed point has been introduced, and this point "keeps track" of the corresponding subvariety. One thinks of this point as the generic point for the subvariety. Furthermore, the sheaf on and the sheaf of polynomial functions on A are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of schemes. Examples The affine scheme is the final object in the category of affine schemes since is the initial object in the category of commutative rings. The affine scheme is scheme theoretic analogue of . From the functor of points perspective, a point can be identified with the evaluation morphism . This fundamental observation allows us to give meaning to other affine schemes. looks topologically like the transverse intersection of two complex planes at a point, although typically this is depicted as a since the only well defined morphisms to are the evaluation morphisms associated with the points . The prime spectrum of a Boolean ring (e.g., a power set ring) is a (Hausdorff) compact space. (M. Hochster) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., a spectral space) if and only if it is quasi-compact, quasi-separated and sober. Non-affine examples Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together. The Projective -Space over a field . This can be easily generalized to any base ring, see Proj construction (in fact, we can define Projective Space for any base scheme). The Projective -Space for is not affine as the global section of is . Affine plane minus the origin. Inside are distinguished open affine subschemes . Their union is the affine plane with the origin taken out. The global sections of are pairs of polynomials on