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 $ϕ : X \to Y$ is an equivalence class of:

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

under the equivalence relation:

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