Theorem 2

Let B be a topological space and A a subspace of B. Make one of the following hypotheses:

(i) A admits a fundamental system of paracompact neighbourhoods;

(ii) B is paracompact and A closed;

(iii) B is metrizable;

(iv) A is compact and two distinct points of A possess disjoint neighbourhoods in B.

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.

Consider the following property (PCV) of the pair $(B,A)$:

(PCV) For every covering $(U_i)_{i\in I}$ of A by open subsets

of B, there exists a neighbourhood V of A and a locally

finite family $(F_j)_{j\in J}$ of closed subsets of V covering V, such that

each of the $F_j$ is contained in one of the $U_i$.

Theorem 2 follows from Lemmas 1 and 3 below.