Let $R$ be a commutative ring and let $S \subset \mathrm{Spec}(R)$ be a subset. Suppose that for each $P \in S$ there exists a Zariski open neighborhood $U$ of $P_P \in \mathrm{Spec}(R_P)$ such that $^aj_{P}(U) \subset S$, where $j_{P}:R \rightarrow R_{P}$ is the canonical morphism; so $S$ can be covered by images of Zariski open subsets under flat morphisms. Is it true that $S$ is open in the flat topology on $\mathrm{Spec}(R)$? If not, is there any useful topology on $\mathrm{Spec}(R)$ where $S$ is open?
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