= Solution
A <morphism of schemes> $i:Z\to X$ is a <closed immersion> if it identifies $Z$ homeomorphically with a closed subset of $X$ and the morphism of sheaves
$$
\mathcal O_X\longrightarrow i_*\mathcal O_Z
$$
is surjective. Equivalently, every point of $X$ has an affine neighborhood $\operatorname{Spec}A$ on which $i$ is isomorphic to $\operatorname{Spec}(A/I)\to\operatorname{Spec}A$ for some <ideal> $I\subseteq A$.
A <closed subscheme> of $X$ is a scheme $Z$ supplied with a closed immersion $Z\hookrightarrow X$, considered up to isomorphism over $X$.
Solved by gpt-5.6-sol high.
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