Vtyjkyuk

The following is exercise 6, chapter 0 from Johnstone's Topos Theory:

Suppose $Frightarrowtail G$ a monomorphism of sheaves in $Shv(X)$ [$X$ being a topological space], and $sigmain G(U)$ for some open $Usubseteq X$. Prove that there is a unique largest open set $Vsubseteq U$ such that $sigma_V$ is in the image of $F(V)$.

Unicity is easy, because given a family of such subsets $V_i_iin I$ then the union $bigcup_I V_i$ would share the same property. However, I can't find a reason for the existence to hold. I tried with the following counterexample:

$X=*$ the singleton space

$Fin Shv(X)$ the constant sheaf sending all opens to the set $a$

$Gin Shv(X)$ the constant sheaf sending all opens to $a,sigma$

Then $Frightarrowtail G$, but a $V$ as above does not exist for any choice of $Usubseteq X$ and taking $sigmain G(U)$.

If $A$ is a sheaf, then $A(varnothing)$ is a one-point set.

So your counterexample is flawed; you give a formula for a constant presheaf, but it is not a sheaf. You have to take the sheafification to get a constant sheaf; e.g. $G(*) = a, sigma $ and $G(varnothing) = * $.

Going back to the generic problem, $sigma|_varnothing$ is the unique element of $G(varnothing)$, which is the image of the unique element of $F(varnothing)$, so there's your existence proof.