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 there exists a sheaf $Q'$ on $(S,\tau)$ for which $Q=f^{-1}Q'$ is the inverse image presheaf.
 
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 $Top^{sep}_Q(S)$? 

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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."