Exercise 8, § 3

Soit B a topological space, let $\mathscr{F}$ be a presheaf on B, let $\widetilde{\mathscr{F}}$ be the associated sheaf and let $\sigma_{\mathscr{F}}:\mathscr{F}\rightarrow \widetilde{\mathscr{F}}$ be the canonical morphism.

a) We say that two elements $s, t\in \mathscr{F}(B)$ are locally equal if every point of B possesses an open neighbourhood U such that $s|U =t|U$. Exercise prove that the relation « $s$ and $t$ are locally equal » is an equivalence relation in $\mathscr{F}(B)$.

b) Exercise prove that, for $s, t\in \mathscr{F}(B)$, one has $\sigma_{\mathscr{F}}(s) =\sigma_{\mathscr{F}}(t)$ if and only if $s$ and $t$ are locally equal.

c) Suppose that B is paracompact and that for every locally finite open covering $(U_i)_{i\in I}$ of B and every family $(s_i)_{i\in I}$, where $s_i\in \mathscr{F}(U_i)$, such that the restrictions to $U_i\cap U_j$ of $s_i$ and $s_j$ coincide, there exists $s\in \mathscr{F}(B)$ such that $s|U_i=s_i$ for every $i$.

Exercise prove that the mapping $\sigma_{\mathscr{F}}(B)$ is surjective.