I am self studying some basic level of algebraic geometry.
Theorem: For any two varieties $X$and $Y$ there is a bijection between:
(i)the set of dominant rational maps from $X$ to $Y$ and
(ii)the set of k-algebra homomorphisms from $K(X)$ to $K(Y)$
I am kind of confused by the proof. It is claimed since any variety has an open cover by affine varieties. Then we can assume $Y$ is affine. It is also claimed that we can assume $Y$ is closed. I did not find this trivial and I don't quite see it.