Proposition 7

Let $p: E\rightarrow B$ be a continuous mapping. In order that the mapping $p$ be étale, it is necessary and sufficient that it be open and that the diagonal $\Delta_E$ of $E\times_BE$ be open in $E\times_BE$.

Suppose first that the mapping $p$ is étale. Then it is open and every point of E possesses a neighbourhood V such that $p|V$ is injective, which amounts to saying that one has $(V\times V)\cap (E\times_BE)\subset \Delta_E$. Hence $\Delta_E$ is an open subset of $E\times_BE$.

Conversely, suppose that $p$ is an open mapping and that $\Delta_E$ is an open subset of $E\times_BE$. Let $x$ be a point of E. Let V be an open neighbourhood of $x$ in E such that $(V\times V)\cap (E\times_BE)$ is contained in $\Delta_E$. Then $p|V$ is injective. By Remark 2, I, p. 29, the mapping $p$ is étale.