Lemma 3

Let B be a topological space and A a subspace of B such that the pair $(B,A)$ satisfies property (PCV) of I, p. 37. Then every continuous section over A of an étale mapping $p: E\rightarrow B$ extends to a continuous section of $p$ over a neighbourhood of A.

Let $p: E\rightarrow B$ be an étale mapping and $s: A\rightarrow E$ a continuous section of $p$ over A. For every point $a$ of A, there exists an open neighbourhood $U_a$ of $a$ and a continuous section $s_a: U_a\rightarrow E$ which coincides with $s$ on $U_a\cap A$ (I, p. 34, Prop. 10 and I, p. 34, Prop. 11). The family $(U_a)_{a\in A}$ is a covering of A by open subsets of B. Let V be a neighbourhood of A in B, let $(F_j)_{j\in J}$ be a locally finite family of closed subsets of V covering V, and let $a: J\rightarrow A$ be a mapping such that $F_j$ is contained in $U_{a(j)}$ for every $j\in J$ (property (PCV)). Let $s_j$ denote the restriction of $s_{a(j)}$ to $F_j$. Let W be the set of points $b\in V$ satisfying $s_j(b) =s_k(b)$ for every pair $(j, k)\in J\times J$ such that $b\in F_j\cap F_k$. One defines a section $s': W\rightarrow E$ by $s'(b) =s_j(b)$ if $b\in F_j$. We have $A\subset W$ and $s'|A =s$. The section $s'$ is continuous by Prop. 4 of I, p. 19.

It remains to prove that W is a neighbourhood of A in B. For every pair $(j, k)\in J\times J$, let $T_{jk}$ denote the set of points $b\in F_j\cap F_k$ such that $s_j(b)=\not s_k(b)$. The set $T_{jk}$ is closed in $F_j\cap F_k$ (Prop. 11, b) of I, p. 34), hence in V, and the sets $T_{jk}$ form a locally finite family of subsets of V. The union T of the family $(T_{jk})$ is therefore a closed set of V (TG, I, p. 6, Prop. 4). Now W is, by definition, the complement of T in V. It is therefore a neighbourhood of A in V and consequently in B.