closed immersion

A morphism of schemes $f:(X,\mathcal{O}_{X})\longrightarrow(Y,\mathcal{O}_{Y})$ is a closed immersion if:

1.

As a map of topological spaces, $f:X\longrightarrow Y$ is a homeomorphism from $X$ into a closed subset of $Y$ , and
2.

the morphism of sheaves $\mathcal{O}_{Y}\longrightarrow\mathcal{O}_{X}$ associated with $f$ is an epimorphism (http://planetmath.org/AbelianCategory) in the category of sheaves.

This notion is the analog of the notion of closed immersion (http://planetmath.org/Immersion) in the category of differential manifolds.