nLab
closed immersion of schemes

A morphism $f:X\to Y$ of schemes is a closed immersion if it induces a homeomorphism of underlying topological spaces (in the Zariski topology) and the comorphism $f^{♯}:{𝒪}_Y\to f_{*}{𝒪}_X$ is an epimorphism of sheaves on $Y$.

More generally, let us consider some category of spaces, i.e. sheaves of sets on $C=\mathrm{Aff}$ equipped with a subcanonical Grothendieck topology. Then a morphism $F\to G$ of spaces is said to be closed immersion if it is representable by a strict monomorphism.