nLab
open immersion of schemes

A morphism $f:X\to Y$ of schemes is an open immersion if the underlying morphism of topological spaces is a homeomorphism onto an open image and the comorphism $f^{♯}:{𝒪}_Y\to f_{*}{𝒪}_X$ is an isomorphism of sheaves when restricted to the image of $f$. In other words, an open immersion is a morphism of schemes which decomposes uniquely into an isomorphism of schemes and the identity inclusion of an open subscheme.