Rational map of varieties

Definition

Let $X$ and $Y$ be varieties over a field $k$ (which we usually assume to be algebraically closed. A rational map $\varphi :X\to Y$ is an equivalence class of:

Pairs $<U,{\varphi }_U>$ where $U$ is an open set in $X$ and ${\varphi }_U$ is a morphism of varieties from $U$ to $Y$

under the equivalence relation:

$<U,{\varphi }_U)\sim <V,{\varphi }_V>$ if and only if ${\varphi }_U$ and ${\varphi }_V$ agree on $U\cap V$.