Intersection of a closed subscheme and an open subscheme of a scheme Let $X$ be a scheme.
Let $Z$ be a closed subscheme of $X$.
Let $U$ be an open subscheme of $X$.
Then $Y = U \cap Z$ is an open subscheme of $Z$.
Can we identify $Y$ with $U\times_X Z$?

Yes. This doesn't have anything to do with closed subscheme. If $p: Z \to X$ is a morphism of schemes and $U \subset X$ is open subscheme, then the fibre product is $p^{-1}(U)$ with open subscheme structure.
