Let $S = (E, p, X)$ be an etale-sheaf of abelian groups, and let $\mathscr F = \Gamma(\Box, S)$. Show: The function $z : X \to E$ defined by $z(x) = 0_x \in E_x$ is a global section.
Some definitions. $S$ is an etale-sheaf if it is a protosheaf (i.e., $p : E \to X$ is a surjective local homeomorphism), The stalks (or fibers) $E_x = p^{-1}(x)$ are abelian groups and addition and inversion are continuous operations. $\Gamma(U, S)$ is the set of all sections $\sigma : U \to E$ continuous and $p \circ \sigma = 1_U$.
I am trying to show that when $z : X \to E$ is defined by $z(x) = 0_x \in E_x$ (The additive identity in the group (stalk) $E_x$) it is a global section, i.e., $z \in \Gamma(X, S)$. To do this I must show $z$ is continuous and $p \circ z = 1_X$. The latter part is trivial, I am stuck on showing continuity.
What I've tried so far: I need to show that $z$ is continuous and have not been able to do this. Let $x \in X$ and let $V$ be an open neighborhood of $z(x) := e$ in $E$. I am trying to find an open set $U$ in $X$ such that $z(U) \subseteq V$. I know that the sheets form a base in $E$ so I can write $V = \bigcup_S V \cap S$. I can map that over to $X$ via $p$ and get an open set. If I could show that $z(p(U)) \subseteq V$ then I would be done, because I know that $p(V)$ is an open neighborhood of $x$. But I don't think this works.
Any ideas? Thanks!
The problem is clearly local: it suffices to find an open cover $\{U_i\}$ of $X$, such that $z|_{U_i}$ is continuous on each $i$. Then $z$ is itself continuous.
Let $y$ be any point of $X$, let $0_y$ be the zero in the fibre above it. By assumption, there is an open neighborhood $W \ni 0_y$ mapping homeomorphically by $p$ onto an open neighborhood $U \ni y$. Let $\sigma: U \to W$ be the inverse homeomorphism. Viewing it as a section, we can take the difference of $\sigma$ with itself fibre by fibre, to get the zero section $0_U : U \to E$. Under the assumption that the group action is continuous, this difference is indeed again a section (this is the essential point). Patching the $0_U$'s together gives the global zero section.