Proposition 12

Let $p: E\rightarrow B$ be an étale mapping. In order that the mapping $p$ be separated, it is necessary and sufficient that for every open subset V of B and every pair $(s, s')$ of continuous sections of $p$ over V, the set of points where $s$ and $s'$ coincide be closed in V.

The condition is necessary (Proposition 11, I, p. 34). Let us prove that it is sufficient. Let $b$ be a point of B, let $x$ and $x'$ be two distinct points of E such that $p(x) =p(x') =b$. Let V be an open neighbourhood of $b$ and $s,s'$ continuous sections of $p$ over V such that $s(V)$ and $s'(V)$ are open neighbourhoods of $x$ and $x'$ respectively (Prop. 9). By hypothesis, the set W of points $x\in V$ such that $s(x)=\not s'(x)$ is open in V, hence in B. The sets $s(W)$ and $s'(W)$ are open neighbourhoods of $x$ and $x'$ respectively; they are disjoint by construction. This proves that the mapping $p$ is separated (I, p. 25, Definition 1).