Let X be a topological space and A be an abelian group. Give A the discrete topology. For any open set U of X, Let $\cal A(U)$ be the group of all continuous aps of U into A. Thus with the usual restriction maps we obtain a sheaf $\cal A$.
So, why for every connected open set U, $\cal A(U)\cong A$ for all U? And, what is the sheafifcation of the presheaf $U \mapsto A$?