Jacobson topology

Remarks

The map $S \mapsto h(k(S))$ satisfies the Kuratowski closure axioms, so it does define a topology. The space $\operatorname{Prim}(R)$ is $T_0$: two distinct primitive ideals $P \ne Q$ have different closures $h(P) \ne h(Q)$. Example: for $R = \Z$ the primitive ideals are the ideals $p\Z$ with $p$ prime. The kernel of an infinite set of them is $0$, so its closure is all of $\operatorname{Prim}(\Z)$; the closed sets are the finite sets and the whole space. Non-example of a closed set: $\set{p\Z : p \text{ an odd prime}}$ is not closed, because its closure contains $2\Z$. For a commutative ring $R$, the primitive ideals are the maximal ideals and the Jacobson topology is the subspace topology from the Zariski topology on the prime spectrum.