Let X be a topological space. Is there such a thing as the constant sheaf on X with value the empty set? That is, a sheaf that assigns to each open set U the set of maps from U to the empty set and to each inclusion the empty map?
I ask because I am trying to show that the projection map from the etale space of a locally constant sheaf on X to X, defined to send a germ at a point x to the point x, is a covering map. It's not clear to me that this map is surjective. That is, why can't there be an x in X such that there are no sections over any neighborhood of x?
The empty sheaf $E$ is exactly as you described it. Note that $E(\emptyset)$ has exactly one element (this is in fact true for all sheaves), and that if $U \neq \emptyset$, we have $E(U) = \emptyset$.
The corresponding etale map is the trivial map from the empty space to $X$. $\DeclareMathOperator{id}{id}$
In general, given a set $A$, the "constant presheaf of $A$" defined by $P(U) = A$ and $P(U \subseteq V) = \id_A$ for all $U, V$ open, and its associated locally constant sheaf $E$, the associated etale space over $X$ will be $p_2 : A \times X \to X$, where $A$ has the discrete topology.