Definition:Restriction of Ringed Space to Open Set

Definition
Let $(X,\OO_X)$ be a ringed space.

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

The restriction of $(X,\OO_X)$ to $U$ is the pair $(X, \OO_X|_U)$.

Here $\OO_X|_U$ is the restriction of $\OO_X$ to $U$.

By Restriction of Sheaf to Open Set is Sheaf $\OO_X|_U$ is a sheaf of rings.

Thus $(X, \OO_X|_U)$ is a ringed space.