Definition 5

Let $p: E\rightarrow B$ be an étale mapping. One says that the mapping $p$ is soft, or that the étale B-space $(E, p)$ is soft, if every continuous section of $p$ over a closed subspace of B extends to a continuous section of $p$ over B.

Let $\mathscr{F}$ be a sheaf on B. One says that $\mathscr{F}$ is a soft sheaf if the associated étale space (I, p. 50, def. 4) is soft.

Let $\mathscr{F}$ be a sheaf on B. The sheaf $\mathscr{F}$ is soft if and only if for every closed Z of B, every open neighbourhood U of Z and every $s\in \mathscr{F}(U)$, there exist $t\in \mathscr{F}(B)$ and an open neighbourhood V of Z contained in U such that $s|V =t|V$.

If $\mathscr{F}$ is a soft sheaf, $\mathscr{F}(B)$ is nonempty: in fact, the unique section of the étale space $E_{\mathscr{F}}$ associated with $\mathscr{F}$ over $\emptyset$ extends to a continuous section of $E_{\mathscr{F}}$ over B.

Let $p: E\rightarrow B$ be an étale mapping and let A be a closed subspace of B. If $p$ is soft, the mapping $p_A:\overset{-1}{p}(A)\rightarrow A$ is soft. Equivalently, if $\mathscr{F}$ is a soft sheaf on B, the sheaf induced on a closed subspace A is soft.