Corollary 2

Let B be a topological space and A a subspace of B. Suppose that the spaces A and B are locally connected and that the pair $(B,A)$ has property (PCV) of I, p. 37. Let E and $E'$ be coverings of B, and let us denote by $p$ and $p'$ their projections. Let

$g:\overset{-1}{p}(A)\rightarrow (^-{p'}^1)(A)$ be an A-morphism ( resp. an A-isomorphism). Then there exists a neighbourhood U of A in B and a U-morphism ( resp. a

U-isomorphism) $f:\overset{-1}{p}(U)\rightarrow (\overset{-1}{p}')(U)$ such that $f_A=g$.

Let us retain the notation of corollary 1. By that same corollary, an A-morphism $g: E_A\rightarrow E'_A$ is identified with a section $s_0$ above A of the étale space $E_{\mathscr{M}}$ associated with $\mathscr{M}$. By the hypothesis made on the pair $(B,A)$ and lemma 3 of I, p. 39, there exists an open neighbourhood U of A in B and a continuous section $s$ of $E_{\mathscr{M}}$ above U extending $s_0$. This section $s$ is identified with a U-morphism $f: E_U\rightarrow E'_U$ extending $f$.

The case where $g$ is an A-isomorphism is treated analogously by considering, in place of the sheaf $\mathscr{M}$, the sheaf $\mathscr{I}$.