Definition:Restriction of Ringed Space to Open Set

Definition
Let $\struct {X, \OO_X}$ be a ringed space.

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

Let $\OO_X {\restriction_U}$ denote the restriction of $\OO_X$ to $U$.

The restriction of $\struct {X, \OO_X}$ to $U$ is the pair $\struct {X, \OO_X {\restriction_U} }$.

Also see

 * Restriction of Sheaf to Open Set is Sheaf, demonstrating $\OO_X {\restriction_U}$ is a sheaf of rings.

Thus $\struct {X, \OO_X {\restriction_U} }$ is a ringed space.