Open Neighborhood contains Affine Open Neighborhood

Theorem

Let $\struct {X, \OO_X}$ be a scheme.

Let $U \subset X$ be an open subset.

Let $x \in U$.

Then there exists an open subset $V \subset U$ with $x \in V$, such that the restriction $\struct {V, \OO_X {\restriction_V}}$ of $\struct {X, \OO_X}$ to $V$ is an affine scheme.

Proof

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