Corollary 4

Let $p: E\rightarrow B$ be an étale and separated mapping. Suppose that E is connected and that $p$ admits a section. Then $p$ is a homeomorphism.

Let $s$ be a section of $p$; since $p$ is étale, the image of $s$ is open in E (Corollary 3); it is also closed, since $p$ is separated (I, p. 27, Prop. 3). Since E is connected, we have $s(B) = E$ and $p$ is a homeomorphism.