This is Exercise II.5 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, it is new to MSE.
The Details:
On p. 66, ibid. . . .
Definition 1: A sheaf of sets $F$ on a topological space $X$ is a functor $F:\mathcal{O}(X)^{{\rm op}}\to\mathbf{Sets}$ such that each open covering $U=\bigcup_iU_i, i\in I$, of open subsets of $U$ of $X$ yields an equaliser diagram
$$ FU\stackrel{e}{\dashrightarrow}\prod_{i\in I}FU_i\overset{p}{\underset{q}{\rightrightarrows}}\prod_{i,j\in I}(U_i\cap U_j),$$
where for $t\in FU,$ $e(t)=\{ t\rvert_{U_i}\mid i\in I\}$ and for a family $t_i\in FU_i$,
$$p\{ t_i\}=\{t_i\rvert_{(U_i\cap U_j)}\}\quad\text{ and }\quad q\{ t_i\}=\{t_j\rvert_{(U_i\cap U_j)}\}.$$
From p. 79, ibid. . . .
For any space $X$, a continuous map $p: Y\to X$ is called a space over $X$ or a bundle over $X$.${}^\dagger$
From p. 82, ibid. . . .
Definition 4: A covering map $p: \stackrel{\sim}{X} \to X$ is a continuous map between topological spaces such that each $x\in X$ has an open neighborhood $U$, with $x\in U \subset X$, for which $p^{-1}U$ is a disjoint union of open sets $U_i$, each of which is mapped homeomorphically onto $U$ by $p$.
On p. 88 ibid. . . .
A bundle $p: E \to X$ is said to be étale (or étale over $X$) when $p$ is a local homeomorphism in the following sense: To each $e\in E$ there is an open set $V$, with $e\in V\subset E$, such that $pV$ is open in $X$ and $p\rvert_V$ is a homeomorphism $V\to pV.$
From the exercise . . .
Definition: A sheaf $F$ on a locally connected space $X$ is locally constant if each point $x\in X$ has a basis of open neighborhoods $\mathcal{N}_x$ such that whenever $U, V \in\mathcal{N}_x$ with $U\subset V$, the restriction $FV\to FU$ is a bijection.
The Question:
Consider a sheaf $F$ on a locally connected space $X$. Prove that $F$ is locally constant iff the associated${}^{\dagger\dagger}$ étale space over $X$ is a covering.
Thoughts:
$(\Rightarrow)$ Let $F$ be a sheaf on a locally connected space $X$. Suppose, further, that $F$ is locally constant. Let $x\in X$. Then there is a basis $\mathcal{N}_x$ such that, for any $U, V\in\mathcal{N}_x$ with $U\subset V$, the restriction $FU\to FV$ is a bijection.
What do I do now?
Looking at the definition of a sheaf, I find myself a little lost here.
$(\Leftarrow)$ I'm completely lost here. I'm not sure I understand the definition of an étale space.
Further Context:
Related questions of mine include the following.
Please help :)
$\dagger$: I'm assuming $Y$ is also a topological space. Am I right?
$\dagger\dagger$: I'm assuming this is the associated bundle as described on page 82, ibid, that, by hypothesis of the question, happens to be étale.
