Exercise 6, § 3

Let $\mathscr{F}$ be a sheaf on a topological space B. We say that $\mathscr{F}$ is flasque if, for every open subset U of B and every section $s\in \mathscr{F}(U)$, there exists a section $\widetilde{s}\in \mathscr{F}(B)$ such that $\widetilde{s}|U =s$.

(a) If B is paracompact, prove that every flasque sheaf is soft.

(b) Suppose that every point $b\in B$ possesses an open neighbourhood U such that the sheaf $\mathscr{F}|U$ induced from $\mathscr{F}$ by restriction to U is flasque. Prove that $\mathscr{F}$ is flasque.

(c) Suppose that $\mathscr{F}$ is flasque. Let A be a topological space and let $u: B\rightarrow A$ be a continuous mapping. Prove that $u_*(\mathscr{F})$ is flasque.

(d) Suppose that B is a metrizable topological space. If $\mathscr{F}$ is a flasque sheaf on B, prove that, for every subset A of B, the sheaf induced on A is still flasque.

(e) Prove that there exist a flasque sheaf $\mathscr{F}'$ and a morphism of sheaves $u:\mathscr{F}\rightarrow \mathscr{F}'$ such that $u(U)$ is injective for every open subset U of B. (Take for the sheaf $\mathscr{F}'$ the sheaf of the not necessarily continuous sections of the canonical mapping from the étalé space $E_{\mathscr{F}}$ into B.)