Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $X$ and $Y$ be affine varieties over $k$, and let $\varphi:X\to Y$ be a set map. Define the graph of $\varphi$ by
Then $\varphi$ is a regular map if and only if $\Gamma_{\varphi}$ is a closed subvariety of $X\times Y$ and the first projection restricts to an isomorphism of affine varieties