Let $F$ be a locally constant sheaf on $X$ and $U$ is an open subset and $F|U$ is a constant sheaf. Let $x\in U$, now let $s,s'$ be two sections from $F(U)$ s.t. $s(x)=s'(x)$, can we say $s=s'$ in a local neighbourhood of $x$?
I think there are two different understandings of "constant"
I know the sections of the constant sheaf $A$ over an open set $U$ may be interpreted as the continuous functions $U\to A$, where $A$ is given the discrete topology. If $U$ is connected, then these locally constant functions are constant.
I feel confused with the example below, where the sections are defined on a connected set however they are not constant, I wonder what the constant means?
If $X$ is locally connected, locally constant sheaves are (up to isomorphism) exactly the sheaves of sections of covering spaces $\pi:Y\to X$.
Such a locally constant sheaf is a constant sheaf if and only the covering $\pi$ is trivial.
So any non trivial covering will give you a non-constant but locally constant sheaf.
The simplest example is the sheaf of sections of the two sheeted non trivial covering $\mathbb C^*\to \mathbb C^*:z\mapsto z^2$ or its restriction to the unit circle $S^1\to S^1: e^{i\theta}\mapsto e^{2i\theta}$
Best Answer
Let $X$ be a topological space and $F$ be a sheave of sets on $X$. As the category of sets on can define stalks of $F$ at points of $X$ and by definition, the stalk $F_x$ of $F$ at an $x\in X$ is the inductive limit of $F(V)$'s indexed over all the open sets $V$ containing $x$, with order relation induced by reverse inclusion. By definition (or universal property) of the inductive limit, two sections over some open $U$ have the same stalk at an $x\in U$ if and only if they coincide on some open neighborhood $V$ of $x$ included in $U$. Thus the answer to you first question is yes and does not depend on $F$ being locally constant or any other hypothesis on $F$.
You are correct about the interpretation of the section of a constant sheaf. By definition a constant sheaf is the sheafification of a constant presheaf, the latter being defined as "all sections over opens are equal to a same set".
Now, about "your" sheaf of sections and covering space, you should look for the definition of espace étalé associated to a preshead, because that is exactly what you call "sheaf of sections" is.