Restriction of Ringed Space to Open Set is Ringed Space

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

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

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

Then $\struct {U, \OO_X {\restriction_U}}$ is a ringed space.

Proof
By Restriction of Sheaf to Open Set is Sheaf $\OO_X {\restriction_U}$ is a sheaf of commutative rings on $U$.

It follows, that $\struct {U, \OO_X {\restriction_U}}$ is a ringed space.