Morphism of varieties

Definition

Let $X$ and $Y$ be two varieties over a field $k$ (which we usually assume to be algebraically closed). A morphism from $X$ to $Y$ is a continuous map $\varphi :X\to Y$ such that:

For any open set $V\subseteq Y$ and any regular function $f:V\to k$ the map $\varphi \circ f:f^{-1}(V)\to k$ is also a regular function.