Proposition 9

In order that a sheaf be locally constant, it is necessary and sufficient that the associated étalé space be a covering.

Let B be a topological space and let $\mathscr{F}$ be a sheaf on B. The sheaf $\mathscr{F}$ is constant if and only if the étalé space $E_{\mathscr{F}}$ is B-isomorphic to a trivial covering $(B\times F$, pr$_1)$, where F is a discrete topological space. If U is an open set of B, the étalé space associated with the induced sheaf $\mathscr{F}|U$ is identified with the induced étalé U-space obtained from $E_{\mathscr{F}}$ over U (cf. I, p. 51). The proposition follows.