Lemma 37.45.1. Let $f : X \to Y$ be a separated, locally quasi-finite morphism with $Y$ affine. Then every finite set of points of $X$ is contained in an open affine of $X$.

**Proof.**
Let $x_1, \ldots , x_ n \in X$. Choose a quasi-compact open $U \subset X$ with $x_ i \in U$. Then $U \to Y$ is quasi-affine by Lemma 37.43.2. Hence there exists an affine open $V \subset U$ containing $x_1, \ldots , x_ n$ by Properties, Lemma 28.29.5.
$\square$

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