# Segre embedding open sets

Given $U,V$ Zariski open sets in $\mathbb{P}^n$ and $\mathbb{P}^m$, under Segre embedding $\phi$, will $\phi (U\times V)$ be open in $\mathbb{P}^{mn+m+n}$? Or if not I guess it will be at least quasiprojective variety? What about $U$ open $V$ closed?

The original question is to prove that given two quasiprojective varieties, then the image under $\phi$ will be quasiprojective variety. I am able to prove it if they are projective, then the image is closed as well, but not sure how to do it for quasiprojective?

Note that you would not expect $\phi(U\times V)$ to be open, since it's dimension is no more than $nm$, while $\mathbb{P}^{nm+n+m}$ is of greater dimension.
$\phi((U\times V)^c)=\phi(\mathbb{P}^n\times\mathbb{P}^m) - \phi(U\times V)$