Corollary 2

Let B be a paracompact space, $\mathscr{F}$ a sheaf on B and $(A_i)_{i\in I}$ a locally finite closed covering of B. In order that the sheaf $\mathscr{F}$ be soft, it is necessary and sufficient that the following condition be satisfied:

For every $i\in I$, every closed subset A of $A_i$, every open set V of B containing A and every element $s$ of $\mathscr{F}(V)$, there exist an open neighbourhood U of $A_i$ in B, an element $t$ of $\mathscr{F}(U)$ and an open neighbourhood W of A in B contained in $U\cap V$ such that $t|W =s|W$.

Suppose that the sheaf $\mathscr{F}$ is soft, and let us show that the condition is satisfied. Let $i\in I$, let A be a closed subset of $A_i$, let V be an open subset of B containing A and let $s$ be an element of $\mathscr{F}(V)$. Put $s_0=\sigma_{\mathscr{F}}(s)$. It is a continuous section over V of the étalé space $E_{\mathscr{F}}$. Since $A_i$ is closed, A is closed in B and $s_0|A$ extends to a section $t_0: B\rightarrow E_{\mathscr{F}}$, by definition of a soft sheaf. The sections $s_0$ and $t_0|V$ of the étalé space over V $E_{\mathscr{F}}\times_BV$ coincide on A, hence on a neighbourhood W of A (I, p. 34, prop. 11, b)).

Conversely, suppose that the condition of the corollary is satisfied. By proposition 6, it is enough to prove that, for every $i\in I$, the sheaf $\mathscr{F}_{A_i}$ is soft. Let A be a closed subset of $A_i$ and let $s_0$ be a section of the etale space of $\mathscr{F}|A_i$ over A, that is, a continuous section over A of the etale space $E_{\mathscr{F}}$. Since A is closed in B and since B is paracompact, $s_0$ extends to a section $s$ over a neighbourhood V of A in B (I, p. 37, th. 2). By hypothesis, there exists an open neighbourhood U of $A_i$ in B and a section $t$ of $E_{\mathscr{F}}$ over U which coincides with $s$ on a neighbourhood of A. The restriction of $t$ to $A_i$ is a section of $\mathscr{F}_{A_i}$ which extends $s_0$. The sheaf $\mathscr{F}_{A_i}$ is therefore soft. By proposition 6, the sheaf $\mathscr{F}$ is soft.