Proposition 13

Let $p: E\rightarrow B$ be an étale and separated mapping and let $s: B\rightarrow E$ be a section of $p$. If the space B is locally path-connected and if the section $s$ is path-continuous, it is continuous.

Let $b$ be a point of B; let us prove that $s$ is continuous at the point $b$. Since $p$ is an étale mapping, there exist a neighbourhood V of $b$ and a continuous local section $s'$ of $p$ defined in V such that $s'(b) =s(b)$ (I, p. 33, prop. 9). Since B is locally path-connected, we may suppose that V is path-connected. For every path $c$ in V, with origin $b$, the mappings $s\circ c$ and $s'\circ c$ are two continuous liftings to B of the mapping $c:\mathbf{I}\rightarrow X$ and we have $s\circ c(0) =s'\circ c(0) =s(b)$. By Corollary 1 of I, p. 34, we have $s\circ c=s'\circ c$ and in particular $s\circ c(1) =s'\circ c(1)$. Since every point of V is the end point of a path in V with origin $b$, the mappings $s$ and $s'$ coincide in V. The mapping $s$ is therefore continuous in V.