Lemma 10.29.3. Let $R$ be a ring. A subset of $\mathop{\mathrm{Spec}}(R)$ is constructible if and only if it can be written as a finite union of subsets of the form $D(f) \cap V(g_1, \ldots , g_ m)$ for $f, g_1, \ldots , g_ m \in R$.

**Proof.**
By Lemma 10.29.1 the subset $D(f)$ and the complement of $V(g_1, \ldots , g_ m)$ are retro-compact open. Hence $D(f) \cap V(g_1, \ldots , g_ m)$ is a constructible subset and so is any finite union of such. Conversely, let $T \subset \mathop{\mathrm{Spec}}(R)$ be constructible. By Topology, Definition 5.15.1, we may assume that $T = U \cap V^ c$, where $U, V \subset \mathop{\mathrm{Spec}}(R)$ are retrocompact open. By Lemma 10.29.1 we may write $U = \bigcup _{i = 1, \ldots , n} D(f_ i)$ and $V = \bigcup _{j = 1, \ldots , m} D(g_ j)$. Then $T = \bigcup _{i = 1, \ldots , n} \big (D(f_ i) \cap V(g_1, \ldots , g_ m)\big )$.
$\square$

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