Assume that $k$ is a field , $X$ a connected topological space, $L$ locally isomorphic to the constant sheaf $k_X$ and $\Gamma(X;L)\neq 0$. I want to show that $L$ is actually isomorphic to the constant sheaf $k_X$.
Morally I understand that if we have a non zero global section of a locally constant sheaf on a connected space, then this section should be a constant function on the space $X$ and $L$ behaves as a constant sheaf. It resembles to me the fact that for projective varieties we don't have non constant global sections of the structure sheaf.
If $\Gamma(X;L)\neq 0$ then it must exist a nonzero morphism inside $Hom_{k_X}(k_X,L)$ but then I'm stuck here, I'd appreciate if anybody could give me an hint.