Corollary 1

Let E and B be topological spaces and let $p: E\rightarrow B$ be a proper and separated mapping. The set U of points $b$ of B such that $p$ is étale at every point of the fibre $E_b$ is open in B. The U-space induced by $(E, p)$ over U is a locally finite covering.

The set F of points of E at which the mapping $p$ is not étale is closed in E (I, p. 29, remark 1). Its image $p(F)$ is closed since the mapping $p$ is proper; the complement U of $p(F)$ is therefore open. The mapping $p_U$ is separated (I, p. 27, prop. 4) and proper (TG, I, p. 72, prop. 3). By construction, the mapping $p_U$ is étale; it therefore satisfies condition (ii) of theorem 1.