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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