**Proof.**
The equivalence of (1), (2), and (3) follows from Lemma 27.2.3. Assume (1), (2), (3). Let $T = X \setminus U$. By Schemes, Lemma 25.12.4 there exists a unique quasi-coherent sheaf of ideals $\mathcal{J}$ cutting out the reduced induced closed subscheme structure on $T$. Note that $\mathcal{J}|_ U = \mathcal{O}_ U$ which is an $\mathcal{O}_ U$-modules of finite type. By Lemma 27.22.2 there exists a quasi-coherent subsheaf $\mathcal{I} \subset \mathcal{J}$ which is of finite type and has the property that $\mathcal{I}|_ U = \mathcal{J}|_ U$. Then $X \setminus U = V(\mathcal{I})$ and we obtain (4). Conversely, if $\mathcal{I}$ is as in (4) and $W = \mathop{\mathrm{Spec}}(R) \subset X$ is an affine open, then $\mathcal{I}|_ W = \widetilde{I}$ for some finitely generated ideal $I \subset R$, see Lemma 27.16.1. It follows that $U \cap W = \mathop{\mathrm{Spec}}(R) \setminus V(I)$ is quasi-compact, see Algebra, Lemma 10.28.1. Hence $U \subset X$ is retrocompact by Lemma 27.2.6.
$\square$

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