The Stacks project

Example 26.3.2. Let $X$ be a locally ringed space. Let $U \subset X$ be an open subset. Let $\mathcal{O}_ U = \mathcal{O}_ X|_ U$ be the restriction of $\mathcal{O}_ X$ to $U$. For $u \in U$ the stalk $\mathcal{O}_{U, u}$ is equal to the stalk $\mathcal{O}_{X, u}$, and hence is a local ring. Thus $(U, \mathcal{O}_ U)$ is a locally ringed space and the morphism $j : (U, \mathcal{O}_ U) \to (X, \mathcal{O}_ X)$ is an open immersion.