I apologize if the following question has already been asked and settled. I couldn't find any thread.

Say, $\mathcal{C} = (Sch/k)$, the category of schemes over $k$ (a field). Let $\mathcal{F} \in \mathcal{C}^{\wedge}$, be an object of $\mathcal{C}^{\wedge}$ - the category of contravariant functors from $\mathcal{C}$ to $(Sets)$. One has the set of points:

$$ |\mathcal{F}| := \lim_{\to} \mathcal{F} (K), $$

the limit taken over fields $K/k$. Given a subfunctor $\mathcal{G} \hookrightarrow \mathcal{F}$ one gets a subset $|\mathcal{G}| \subset |\mathcal{F}|$ (ie. a *canonical* map from $|\mathcal{G}| \to |\mathcal{F}|$ that is injective). In particular, $|\mathcal{U}|$ for the open subfunctors $\mathcal{U} \hookrightarrow \mathcal{F}$ form a topology on $|\mathcal{F}|$.

Question: Given a closed subset $Z \subset |\mathcal{F}|$ does there exist a closed subfunctor (possibly non-unique) $\mathcal{Z} \hookrightarrow \mathcal{F}$ so that $Z = |\mathcal{Z}|$ (as subsets of $|\mathcal{F}|$)?

In some sense, are *open subfunctors* and *closed subfunctors* really "complimentary"?