ag.algebraic geometry - Homeomorphism onto a closed subset of a scheme that isn't a closed immersion

Yes, for example if $K subset L$ is an inclusion fields, then the induced map
Spec $L to$ Spec $K$ is a homeomorphism (both source and target are single points),
but the induced map on sheaves is the given inclusion of $K$ into $L$, which is
surjective only if $K = L$.

For another example, let $X'to Y$ be a closed immersion of schemes over ${bar{mathbb F}}_p$, and let $X to X'$ be the relative Frobenius morphism.
Then $Xto X'$ is a homeomorphism on underlying topological spaces but is not an isomorphism of schemes, and so the composite $Xto Y$ is a closed embedding on underlying spaces but not a closed immersion of schemes.

As one last example, let $X'$ be the cuspidal cubic given by $y^2 = x^3$ in the affine
plane $Y$ (over $mathbb C$, say), and let $X$ be the normalization of $X'$ (which is just
the affine line). Then $X to X'$ is a homeomorphism on underlying spaces, but is not
an isomorphism of schemes. The composite $X to Y$ is thus not a closed immersion,
but induces a closed embedding of underlying topological spaces.

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