Restriction of Ringed Space to Open Set is Ringed Space
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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.
$\blacksquare$