Example 2

Let B be a topological space and $(E, p)$ a B-space. Let $\mathscr{F}$ be the sheaf on B of continuous sections of $(E, p)$. The mapping $(U, s, x)\mapsto s(x)$ of $X_{\mathscr{F}}$ into E is compatible with the equivalence relation $R_{\mathscr{F}}$. The mapping $e: E_{\mathscr{F}}\rightarrow E$ deduced from it by passing to the quotient is a B-morphism; the B-morphism $e$ is said to be canonical. The image of $e$ is the union of the images of the continuous sections of $p$ over the open sets of B. The mapping $e$ is therefore surjective if $p$ is etale (I, p. 33, prop. 9). On the other hand, the mapping $e$ is injective if and only if for every open set U of B and every pair $(s, s')$ of continuous sections of $p$ on U, the set of points $x\in U$ such that $s(x) =s'(x)$ is open; this is in particular the case if $p$ is etale (I, p. 34, prop. 11, b)). Consequently, if $(E, p)$ is an etale B-space, the mapping $e$ is a B-isomorphism.