proving an isomorphism of two open sets (Hartshorne Corollary I 4.5)

proving an isomorphism of two open sets (Hartshorne Corollary I 4.5)

Let $X,Y$ be birational varieties and $\phi: X \rightarrow Y, \psi: Y
\rightarrow X$ mutually inverse rational dominant maps. Let $\phi$ be
represented by $(U,\phi_U)$ and $\psi$ by $(V,\psi_V)$. Then Hartshorne in
Corollary I 4.5 says that the open sets $\phi_U^{-1}(\psi_V^{-1}(U))
\subset U$ and $\psi_V^{-1}(\phi_U^{-1}(V)) \subset V$ are isomorphic via
$\phi$ and $\psi$ respectively.
I am wondering if he means that they are isomorphic as topological spaces
or as varieties. Moreover, how can we actually see this isomorphism? It
seems to me a little bit confusing. Here is the closest i got:
$\phi_U(\phi_U^{-1}(\psi_V^{-1}(U)) \subset \psi_V^{-1}(U) =
\psi_V^{-1}(\phi_U^{-1}(Y)) = \psi_V^{-1}(\phi_U^{-1}(V))^c$, where $^c$
denotes closure.