Given a set $S$, let $Top(S)$ denote the partially ordered set (poset) of topologies on $S$, ordered by fineness, so the discrete topology, $Disc(S)$, is maximal.

Suppose that $Q$ is a presheaf on $Disc(S)$, that $\tau$ is a topology on $S$, and that $\epsilon\colon Disc(S)\to (S,\tau)$ is the canonical map. I'll say that $Q$ is *a sheaf with respect to $\tau$* if the direct image presheaf $\epsilon_*(Q)$ is a sheaf.

For a given set $S$ and presheaf $Q$, let $$Top_{Q}(S)\subseteq Top(S)$$ denote the subposet of topologies on $S$ for which $Q$ is a sheaf.

For example, if $1$ is the terminal presheaf then $Top_{1}(S)=Top(S)$, i.e., every topology makes $1$ a sheaf. In general $Top_{Q}(S)$ may be empty, e.g., if $Q$ assignes a non-terminal set $Q(\emptyset)\not\cong\{*\}$ to the emptyset $\emptyset\subseteq S$.

**Question 1:** In general, what can one say about the poset $Top_Q(S)$? For example, is it closed under binary meets or joins in $Top(S)$?

Let $Top^{sep}_Q(S)$ denote the poset of $S$-topologies on which $Q$ is a separated presheaf. Recall that a presheaf on a space $X$ is separated if every matching family of sections on a cover extends to at most one section on their union. This condition is less stringent than the sheaf condition, which replaces *at most one* with *exactly one*. In general, we have
$$Top_Q(S)\subseteq Top^{sep}_Q(S).$$
For example the initial presheaf $0$ on $Disc(S)$ is a separated presheaf but not a sheaf, so $Top_0(S)\subsetneq Top^{sep}_0(S)$.

**Question 2:** What can one say about the poset $Top^{sep}_Q(S)$?

Edit provenance: I added the second paragraph to address a question of Zhen Lin, which pointed out an ambiguity in the phrase "topologies on $S$ for which $Q$ is a sheaf." Later I added even more detail about that notion, to coincide with clarifications made in the comments. Even later, I realized that the notion I had originally chosen (that $Q$ was the inverse image presheaf $\epsilon^{-1}(Q')$ of some sheaf $Q'$ on $(S,\tau)$) was a bit pathological, so I weakened the condition to say that $\epsilon_*Q$ is a sheaf.

all$V \subseteq S$. $\endgroup$ – Zhen Lin Apr 14 '14 at 22:57