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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