A topology where open sets are huge and closed sets are rare — and why that's exactly right
Once we have the set Spec(R), we need a topology on it — a notion of which subsets are "open" and "closed." The Zariski topology declares the closed sets to be the vanishing loci V(I) = {𝔭 ∈ Spec(R) | I ⊆ 𝔭} for ideals I of R. This is a coarse topology: open sets are large, points are rarely separated, and the whole space is far from Hausdorff. But this coarseness is a feature, not a bug — it captures exactly the algebraic information we need.
The basic open sets are D(f) = {𝔭 ∈ Spec(R) | f ∉ 𝔭} for elements f ∈ R. These form a basis for the topology and are the building blocks of the structure sheaf. If you remember the Zariski topology on affine varieties from the AG module, this is the same idea, promoted to work on any ring — and the non-Hausdorff nature now has a clear algebraic meaning through the specialization order.
For an ideal I ⊆ R, define V(I) = {𝔭 ∈ Spec(R) | I ⊆ 𝔭}. The closure of a single point 𝔭 is V(𝔭) — the set of all primes containing 𝔭. For a maximal ideal 𝔪, V(𝔪) = {𝔪} is a single point, so maximal ideals are closed. But V((0)) in an integral domain is all of Spec(R), which means the generic point (0) is dense — its closure is the entire space.
The basic open sets D(f) = Spec(R) ∖ V(f) form a basis for the topology. Crucially, D(f) is itself homeomorphic to Spec(Rf), where Rfis the localization of R at f. This means every basic open set is again the spectrum of a ring — the topology and the algebra are perfectly intertwined.
The Zariski topology is almost never Hausdorff. Two distinct points 𝔭 and 𝔮 can be separated by open sets only if neither contains the other. When 𝔭 ⊆ 𝔮, we say 𝔭 specializes to 𝔮 — meaning 𝔮 lies in the closure of {𝔭}. The generic point of an irreducible scheme specializes to every other point, reflecting the fact that a "general" element of a variety can be deformed into any specific point.
This non-Hausdorff behavior looks strange from the perspective of point-set topology, but it is exactly right for algebraic purposes. The specialization partial order encodes the inclusion relations among irreducible closed subsets. In Spec(ℤ), the generic point (0) specializes to every (p), capturing the fact that a "generic integer" can be reduced modulo any prime.