I am a newcomer to sheaf theory. A possible user only. It came up, driven by a particular certain naturality of restriction.
If $\mathcal F$ is a sheaf of sets over $\mathcal X$ then one can define the notion of a maximal section: a section $s\in\mathcal F(U)$, $U$ connected, such that, if $V\supseteq U$, $V$ is connected, and $t\in \mathcal F(V)$ and $t|U=s$ then $U=V$. I am interested in this notion, and particularly, given an element $\gamma$ of the etale space, whether or not there is a unique maximal section $s\in\mathcal F(U)$ such that $\gamma=[s,U,x]$ where $[s,U,x]$ is the usual germ in the etale space. This is a worthwhile question only if the etale space is Hausdorff, in which case two maximal sections are equal on the connected components of the intersection of their domains. If there is only one component then they are equal because an extension would be obtained by gluing them, contradicting maximality, but there may be multiple such components.
I have three simple examples. Given the sheaf of smooth functions on $\mathbb R$ with derivative $1$ the answer is yes, obviously. Given the same sheaf where the functions satisfy $s'=1+s^2$ the answer is yes, but the domains are all open intervals of length $\pi$. Given the smooth functions on the circle $S^1$ which satisfy $ds/d\theta=1$ ($d/d\theta$ is the obvious vector field) the answer is no, because the maximal sections, say with $s(1,0)=0$, are all linear in $\theta$ so to speak but the domains can be $S^1$ minus any point except $(0,1)$ and the point can vary.
Does anyone know of a relevant reference, or maybe an expert's tip? I've reviewed quite a lot and I am pretty comfortable now with sheaves in general, but I am not so confident or clear on how to approach this. The maximal sections seem far away from the sheaf cohomology that I read.