The answer to your question is pleasantly general and simple.
Given a completely general scheme $X$ (no noetherian, separation, ...hypothesis) and an arbitrary closed subspace $F\subset |X|$ of its underlying topological space, there is a unique closed reduced subscheme $Y\subset X$ whose underlying set is $|Y|=F$. Here is the proof:
i) If $X=Spec A$ is affine, $Y$ is given by the ideal $I=\sqrt J$, with $J=\bigcap_{x\in F} j_x \;$
[as usual, for $x\in SpecA, j_x \subset A$ denotes the ideal corresponding to the point $x$],
ii) If $X$ is not affine, the reduced scheme $Y=V_{sch}(\mathcal I)$ is obtained by the unique ideal sheaf $\mathcal I\subset \mathcal O_X$ restricting on each open affine $U=Spec A$ to the ideal sheaf $\tilde I$ associated to the $I$ above.
Reference EGA 1, Chap.1 , §5, Proposition (5.2.1)