Given a commutative ring $R$, it seems to me that there are two obvious topologies one can put on the set $\operatorname{Spec}R$ of its prime ideals. First is the Zariski topology, with closed sets like \begin{equation} V(I) = \{ \mathfrak{p} \in \operatorname{Spec}R \, | \, I \subseteq \mathfrak{p} \}, \end{equation} where $I \subseteq R$ is any ideal. Second is the poset topology given by inclusion of primes, which has a basis given by sets like $V(\mathfrak{q})$ for $\mathfrak{q} \in \operatorname{Spec}R$. These topologies produce compact spaces for any $R$, and moreover both are contractible when we take $R$ to be an integral domain; the generic point $(0)$ even makes $\operatorname{Spec}(R)$ into a non-Hausdorff cone in the second case.
But it seems to me that these topologies need not be the same in general—the poset topology might have fewer open sets. However, in light of the formula $V(IJ) = V(I) \cup V(J)$, I suspect the topologies coincide for Dedekind domains. I'm also not certain in general what ideals can be constructed by summing over primes (that is, what we might get by allowing arbitrary intersections).
Does anyone have any light they could shed on the matter?