Lemma 26.9.2. Let $X$ be a scheme. Let $j : U \to X$ be an open immersion of locally ringed spaces. Then $U$ is a scheme. In particular, any open subspace of $X$ is a scheme.

**Proof.**
Let $U \subset X$. Let $u \in U$. Pick an affine open neighbourhood $u \in V \subset X$. Because standard opens of $V$ form a basis of the topology on $V$ we see that there exists a $f\in \mathcal{O}_ V(V)$ such that $u \in D(f) \subset U$. And $D(f)$ is an affine scheme by Lemma 26.6.6. This proves that every point of $U$ has an open neighbourhood which is affine.
$\square$

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